From jean-luc.manguin at unicaen.fr  Tue Jun  2 16:51:34 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Tue, 2 Jun 2020 16:51:34 +0200 (CEST)
Subject: [seqfan] A property of numbers in A171640
Message-ID: <1606652363.12731015.1591109494544.JavaMail.zimbra@unicaen.fr>

Hello, 

I would like to add a comment to the serie [ https://oeis.org/A171640 | A171640 ] 
But as I have 3 pending drafts I can't edit this sequence... 
Anyway I post it here. 
If a(n) is in A171240, then the number N = 6 x a(n) x (a(n) -1) is a square number. 
Moreover, if we consider the number P = 2 x (a(n) - 1) + sqrt(N) then : 

P x P + (P+1)x(P+1) + (P+2)x(P+2) = (P + a(n))x(P + a(n)) + (P + a(n) +1)x(P + a(n) +1) 

The first term a(1)=3 gives P=10 
and 10*10 + 11*11 + 12*12 = 13*13 + 14*14 = 365 

Thank you for your attention. 
Best regards. 

JL Manguin 



From mlb at well.com  Tue Jun  2 19:12:28 2020
From: mlb at well.com (Marc LeBrun)
Date: Tue, 2 Jun 2020 10:12:28 -0700
Subject: [seqfan] Re: A property of numbers in A171640
In-Reply-To: <1606652363.12731015.1591109494544.JavaMail.zimbra@unicaen.fr>
References: <1606652363.12731015.1591109494544.JavaMail.zimbra@unicaen.fr>
Message-ID: <026AEAD4-6BF2-4CDB-9039-B5BA06191A02@well.com>

>= Jean-Luc Manguin <jean-luc.manguin at unicaen.fr>
> If a(n) is in A171240...

Did you mean A171640 here?



From sven-h.simon at gmx.de  Wed Jun  3 21:29:31 2020
From: sven-h.simon at gmx.de (Sven Simon)
Date: Wed, 3 Jun 2020 21:29:31 +0200
Subject: [seqfan] Re: Websites on OEIS server
In-Reply-To: <CAAOnSgTxEOw_tG19WTtU85t2wTJ6-42BSFy=hr4ADCz71fiWgw@mail.gmail.com>
References: <000001d634fb$26ab3b40$7401b1c0$@gmx.de>
 <CAAOnSgSCDPvDHLSesk5WMScG78d+ebASvWfJwsPqtg8GdAD1mw@mail.gmail.com>
 <000001d63662$e8225f70$b8671e50$@gmx.de>
 <CAAOnSgTxEOw_tG19WTtU85t2wTJ6-42BSFy=hr4ADCz71fiWgw@mail.gmail.com>
Message-ID: <002d01d639dd$4ef2d0e0$ecd872a0$@gmx.de>

Hello Neil,
There was another email to the topic which did not make it into our emails.
I was at another location for a short period and did send the email another
way. This is it:
Hello Neil,
unforunately I do not have any experience in programming web pages. I think
it would be to much work to migrate any of the links included in the web
page and some of them are dead already either. If you only take the text it
is less work I estimate and one has more motivation to use the original
page. It is not our intention to remove the original page.
What I could do as a workaround is making screenshots of graphics and
interesting tables and we could place the jpg-files on the OEIS server and
link them to the sequence.
 
Sven

-----Urspr?ngliche Nachricht-----
Von: SeqFan <seqfan-bounces at list.seqfan.eu> Im Auftrag von Neil Sloane
Gesendet: Samstag, 30. Mai 2020 21:22
An: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc: Achim Flammenkamp <achim at uni-bielefeld.de>
Betreff: [seqfan] Re: Websites on OEIS server

Sven,  Thank you for that reply.  I will have to study this problem more
carefully, but I can't do it right now.  Very very busy.

I will try to do it soon.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sat, May 30, 2020 at 5:15 AM Sven Simon <sven-h.simon at gmx.de> wrote:

> Hello Neil,
> After a closer look on Achim Flammenkamp's webpage on multiperfect 
> numbers I think it is best to give you all that work of putting it to 
> the OEIS server. It does not make much sense if I make a Html file  of 
> it, and there will be this or that missing or it is not in the right 
> format while you can easily access it and make necessary changes. 
> Almost certainly not all the links included are suitable to have on 
> OEIS server. Might be we do not need any of the links at all. For 
> example there is a search function to search in the numbers list, that 
> would be a lot of work to migrate and it is not so urgent if you have 
> the complete data file anyway. Of course parts of the webpage describing
the search function would be not necessary then too.
>
> The page itself without further links is useful already in my opinion, 
> it has some interesting lists and graphics.
>
> The link to his multiperfect numbers page is
>
> http://wwwhomes.uni-bielefeld.de/achim/mpn.html
>
> OEIS sequence is A091443.
>
> The link to his page is already at the OEIS  sequence.
> His permission is to make a copy of his multiperfect numbers page 
> (link
> above) to place it on OEIS server, name it so it is clear it is a copy 
> of his page and add the date of the copy in the name. The link already 
> at the sequence shall remain, so when there are changes on his page at 
> Uni Bielefeld, we get these too. And one can use his additional 
> functionality on the original webpage.
>
> For those who speak German, that is his permission statement:
> Machen Sie ihre physische Kopie, stellen Sie diese oeffentlich, ABER 
> packen sie einfach eine Anmerkung/Note das dies eine Kpoie der 
> LINK-auf-original URL vom 2020-XX-YY ist.
> Dann kann jeder der mag auf den Link klicken und wird, falls sie oder 
> gar eine neuere Version dort ist diese sehen.
>
> Thanks
> Sven
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Seqfan Mailing list - http://list.seqfan.eu/



From alonso.delarte at gmail.com  Fri Jun  5 03:56:02 2020
From: alonso.delarte at gmail.com (Alonso Del Arte)
Date: Thu, 4 Jun 2020 21:56:02 -0400
Subject: [seqfan] Haverbeke's 1 plus 5 or times 3 problem
Message-ID: <CAGyGvfVCZ-9bRWaT4PkDuRUTzo7Db0h=G-6_jBd4-h-CT0DDbg@mail.gmail.com>

To illustrate recursive functions in *Eloquent JavaScript*, Marijn
Haverbeke posits the following problem: start from 1, then either add 5 or
multiply by 3. Repeat one or the other or alternate in whatever pattern you
want. Some numbers, like 13, can be reached in at least one way. Others,
like 15, are unreachable. Write a JavaScript function to find a solution
for a given number; it doesn't have to be the "best" solution, it simply
has to be *a* solution, if it exists for the particular number.

Maybe Haverbeke's the first to connect this mathematical problem to
JavaScript, but I doubt he's the one who first formulated it. Although it's
also possible that he remembered a similar problem and intentionally or
unintentionally changed it to suit his purpose. Though I doubt anyone
before me has explored the variant starting with ?1, which I'm not ready to
say anything more about at this time.

As usual for problems of this sort, the first reference to consult is the
OEIS. My searches got lots of results, but in this instance I suspect they
were often irrelevant matches, and a demonstration of the law of small
numbers.

Haverbeke bibliographic citation and link to Scastie snippet currently at
https://oeis.org/draft/A335365

My question to y'all: have you ever heard of this problem before, or a
similar problem?

Al

-- 
Alonso del Arte
Author at SmashWords.com
<https://www.smashwords.com/profile/view/AlonsoDelarte>
Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>


From njasloane at gmail.com  Fri Jun  5 04:40:35 2020
From: njasloane at gmail.com (Neil Sloane)
Date: Thu, 4 Jun 2020 22:40:35 -0400
Subject: [seqfan] Re: Haverbeke's 1 plus 5 or times 3 problem
In-Reply-To: <CAGyGvfVCZ-9bRWaT4PkDuRUTzo7Db0h=G-6_jBd4-h-CT0DDbg@mail.gmail.com>
References: <CAGyGvfVCZ-9bRWaT4PkDuRUTzo7Db0h=G-6_jBd4-h-CT0DDbg@mail.gmail.com>
Message-ID: <CAAOnSgTdvkqQiAuseTEB93xNZewdHXrih7g4WsNGNG5em2ij0Q@mail.gmail.com>

This is a type of sequence where we already have a number of examples
(although not that one). For example:

%N A121538 Increasing sequence: "if n appears then a*n+b doesn't", case
a(1)=1, a=2, b=1.
%C A121539 Equivalently, increasing sequence defined by: "if n appears
a*n+b does not", case a(1)=0, a=2, b=1.
%N A121540 Increasing sequence: "if n appears a*n+b does not", case a(1)=3,
a=2, b=1.
%N A121541 Increasing sequence: "if n appears a*n+b does not", case a(1)=4,
a=2, b=1.
%N A121542 Increasing sequence: "if n appears a*n+b does not", case a(1)=5,
a=2, b=1.
%C A003159 If n appears then 2n does not.
%N A005658 If n appears so do 2n, 3n+2, 6n+3.
%N A005659 If n appears so do 2n-2 and 3n-3.
%N A005660 If n appears so do 2n+2 and 3n+3.
%N A005662 If n appears then so do 2n+2 and 3n+3.
%C A022342 If n appears, n + (rank of n) does not (10 is the 7th term in
the sequence but 10 + 7 = 17 is not a term of the sequence). - _Benoit
Cloitre_, Jun 18 2002
%C A028260 If n appears, p*n does not (p primes). - _Philippe Del?ham_, Jun
10 2006
%C A036668 If n appears then 2n and 3n do not. - _Benoit Cloitre_, Jun 13
2002
%N A121537 If n appears then 2n, 3n and 4n do not.
%N A121543 "If n appears then n-th prime doesn't", with a(1)=1.

Besides your A335365, could you also submit "If n appears then so do n+5
and 3*n",  for completeness, even though it is very dense ?
I guess it begins
1, 3, 6, 8, 9, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, ...


On Thu, Jun 4, 2020 at 9:56 PM Alonso Del Arte <alonso.delarte at gmail.com>
wrote:

> To illustrate recursive functions in *Eloquent JavaScript*, Marijn
> Haverbeke posits the following problem: start from 1, then either add 5 or
> multiply by 3. Repeat one or the other or alternate in whatever pattern you
> want. Some numbers, like 13, can be reached in at least one way. Others,
> like 15, are unreachable. Write a JavaScript function to find a solution
> for a given number; it doesn't have to be the "best" solution, it simply
> has to be *a* solution, if it exists for the particular number.
>
> Maybe Haverbeke's the first to connect this mathematical problem to
> JavaScript, but I doubt he's the one who first formulated it. Although it's
> also possible that he remembered a similar problem and intentionally or
> unintentionally changed it to suit his purpose. Though I doubt anyone
> before me has explored the variant starting with ?1, which I'm not ready to
> say anything more about at this time.
>
> As usual for problems of this sort, the first reference to consult is the
> OEIS. My searches got lots of results, but in this instance I suspect they
> were often irrelevant matches, and a demonstration of the law of small
> numbers.
>
> Haverbeke bibliographic citation and link to Scastie snippet currently at
> https://oeis.org/draft/A335365
>
> My question to y'all: have you ever heard of this problem before, or a
> similar problem?
>
> Al
>
> --
> Alonso del Arte
> Author at SmashWords.com
> <https://www.smashwords.com/profile/view/AlonsoDelarte>
> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From jean-luc.manguin at unicaen.fr  Fri Jun  5 14:06:46 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Fri, 5 Jun 2020 14:06:46 +0200 (CEST)
Subject: [seqfan] Two questions about A287151 and A292357
Message-ID: <861407980.15411096.1591358806081.JavaMail.zimbra@unicaen.fr>

Hello, 

After considering A287151 and A292357, I have two questions : 

1) These series count "fixed" structures, I mean they do not take in account possible rotations or symmetries ; I did not manage to find their equivalent for "one-sided" structures nor for "free" structures (except A268371, which is much less developped). Does anyone know if this kind of sequence exist in OEIS ? 

2) These series count the total of structures in a m x n rectangle, but I would like to find detailed information about the number of cells (or elements) in the structures, as in A054252 for the squares, and A226048 for 2 x n rectangles, etc... ; my question is similar to the first one : does anyone know if this kind of sequence exist in OEIS ? 

In addition : I have data, but I try to search sequences starting from my data, and I got nothing ; anyway I prefer to ask first because they might exist in a different format (table read by antidiagonal, for instance). 

Thank you by advance for the answers. 
Best regards. 

JL Manguin 



From wouter.meeussen at telenet.be  Sun Jun  7 20:16:16 2020
From: wouter.meeussen at telenet.be (Wouter Meeussen)
Date: Sun, 7 Jun 2020 20:16:16 +0200
Subject: [seqfan] interesting idea, weird ramifications, as yet unsolved
Message-ID: <D9FBE905A7AB491997B0748B761F565B@Wou>

?Pairwise combinations of distinct elements?
see https://math.stackexchange.com/questions/3708151

my feeling is that Algebraic Combinatorics should clarify this, reducing it to a well known pattern.
I tried but failed dismally.

It boils down to a list of monomials X_(k+1) generated by

(X_k ** X_k  - X_k . X_k )/2 where ?**? represents the outer product.

The author (?Cesare?) only asks for a very limited property of these multinomials, but there is a lot of other ?enumerative combinatorics? going on here.

Does anyone recognise the patten?

Wouter.


From jmachacek.math at gmail.com  Mon Jun  8 04:23:28 2020
From: jmachacek.math at gmail.com (John Machacek)
Date: Sun, 7 Jun 2020 22:23:28 -0400
Subject: [seqfan] Re: questions about walks in the plane
In-Reply-To: <CA+Tsd+34_u-R_yxYFB78W0bLF6vfg+MEk4+gsLNaA4r9CULnPA@mail.gmail.com>
References: <CAAOnSgQCWJN1z9ctomfAJsrBbMAZNetrSXXCxs28RrnSEL01vA@mail.gmail.com>
 <MN2PR02MB6079AF46BEA2D848C28B11AAD98C0@MN2PR02MB6079.namprd02.prod.outlook.com>
 <CA+Tsd+3_XgU-LPs6k=+=yRAF4uaSWx1M+qnTET8+6+kdMLR1WA@mail.gmail.com>
 <CA+Tsd+34_u-R_yxYFB78W0bLF6vfg+MEk4+gsLNaA4r9CULnPA@mail.gmail.com>
Message-ID: <CA+Tsd+3xQbPteeZJgwTQd3EDCwYQFoLw7nB5ORu5p8WieF+1xQ@mail.gmail.com>

Hello,

I can explain the following

%C A085363 Apparently, the number of 2-D directed walks of semilength n
starting at (0,0) and ending on the X-axis using steps NE, SE, NW and SW
avoiding adjacent NW/SE and adjacent NE/SW. - _David Scambler_, Jun 20 2013

I come up with a new formula. I wasn't able to count the walks directly
with anything in A085363. Let s(n) be the number of such 2-D directed walks
of semilength n satisfying the conditions. I claim

s(n) = Sum_{k=1..n} [2^(2*n-2*k+2)*binomial(n-1,k-1)^2 +
2^(2*n-2*k+3)*binomial(n-1,k-1)*binomial(n-1,k-2)].

Then using Zeilberger's Algorithm we find that n*s(n) = (10*n-6)*s(n-1) -
(9*n-18)*s(n-2) which is the same recursion as for a(n) in A085363. We can
check the initial values match. Therefore s(n) = a(n) = A085363(n).

To see why the s(n) formula is correct let's look at an example.

Let's look at the term 2^(2n-2k+2)*binomial(n-1,k-1)*(binomial(n-1,k-1)
with n=4 and k = 2. Here choose two integer compositions of n each with k
parts. For example, (3,1) and (2,2). We then interleave to make a word
UUUDDUDD where U means up and D means down.

UUU (D)D (U) (D)D

Now with the exception of the letters in parentheses each U can be replaced
with NE or NW and similarly each D can be replaced with SE or SW. The
letters in parentheses are forced. There are 2^5 = 2^(8 - 4 + 1) choices.
The extra factor of 2 comes from switching roles of U and D (i.e.
reflecting path over axis).

The term 2^(2*n-2*k+3)*binomial(n-1,k-1)*binomial(n-1,k-2) is similar
except you have compositions like (2,2) and (4) so a word like UUDDDDUU.

We also get two hypergeometric evaluations.

s(n) = 4^n 3F2(-n, 1+n, 1-n; 1, n; 1/4)
s(n) = (8n - 4)* 3F2(2-2n, 1-n, 1-n; 1-2n, 2; 4)

Best,
John Machacek

On Sun, May 31, 2020 at 12:10 PM John Machacek <jmachacek.math at gmail.com>
wrote:

> ...sorry typo in my previous email on the initial conditions. Of course
> a(2) = 1 since the empty walk is the unique walk of length zero.
>
> On Sun, May 31, 2020 at 12:08 PM John Machacek <jmachacek.math at gmail.com>
> wrote:
>
>> Hello,
>>
>> For A001630 the walks are "ternary" words in {-1, +1, +3}, but the
>> "length" is sum of the absolute values (as opposed to length as a word in
>> {-1, +1, +3}).
>>
>> So, a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) since
>>
>> a(n-1) accounts for walks ending with (+1)
>> a(n-2) accounts for walks ending with (+1, -1)
>> a(n-3) accounts for walks ending with (+3)
>> a(n-4) accounts for walks ending with (+3, -1).
>>
>> Then we check the initial conditions also work
>>
>> a(2) = 0: (empty)
>> a(3) = 2: (+1), (-1)
>> a(4) = 3: (+1,+1), (+1,-1), (-1,+1)
>> a(5) = 6: (+1,+1,+1), (+1,+1,-1), (+1,-1,+1), (-1,+1,+1), (-1,+1,-1), (+3)
>>
>> I don't currently have anything to say about the 2-D walks. But I'll try
>> to think about them...
>>
>> Best,
>> John Machacek
>>
>>
>> On Sun, May 31, 2020 at 3:25 AM Nacin, David <NACIND at wpunj.edu> wrote:
>>
>>> Though the last three questions are clear, I'm confused on the first.
>>> If we are talking one-D walks using +1,-1, +3 with no consecutive -1's then
>>> there must be some other restriction as well, otherwise the sequence would
>>> just contain A028859<https://oeis.org/A028859>
>>> 1,3,8,22,60,164,448,1224,... .  (It also wouldn't matter what the numbers
>>> themselves were, only that one of the numbers can't be repeated
>>> consecutively.)  What am I missing?  Does the walk have to end at a certain
>>> value?
>>>
>>> -David
>>>
>>> ________________________________
>>> From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of Neil Sloane <
>>> njasloane at gmail.com>
>>> Sent: Thursday, May 28, 2020 12:26 PM
>>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>> Subject: [seqfan] questions about walks in the plane
>>>
>>> An old friend (Kees Immink) asked me about the conjecture of David
>>> Scambler
>>> in A085363.  In fact there are 4 assertions/conjectures in the OEIS of
>>> this
>>> type: (the first is only a one-D walk)
>>>
>>> %C A001630 Apparently for n>=2 the number of 1-D walks of length n-2
>>> using
>>> steps +1, +3 and -1, avoiding consecutive -1 steps. - _David Scambler_,
>>> Jul
>>> 15 2013
>>>
>>> %C A084768 Number of directed 2-D walks of length 2n starting at (0,0)
>>> and
>>> ending on the X-axis using steps NE, SE, NE, SW and avoiding NE followed
>>> by
>>> SE. - _David Scambler_, Jun 24 2013
>>>
>>> %C A085363 Apparently, the number of 2-D directed walks of semilength n
>>> starting at (0,0) and ending on the X-axis using steps NE, SE, NW and SW
>>> avoiding adjacent NW/SE and adjacent NE/SW. - _David Scambler_, Jun 20
>>> 2013
>>>
>>> %C A101500 Directed 2-D walks with n steps starting at (0,0) and ending
>>> on
>>> the X-axis using steps N,S,E,W and avoiding N followed by S. - _David
>>> Scambler_, Jun 24 2013
>>>
>>> I know we have several experts here - could someone help and provide
>>> proofs?
>>>
>>> The third question is the following:
>>> Let a(n) = the number of 2-D directed walks of semilength n starting at
>>> (0,0) and ending on the X-axis using steps NE, SE, NW and SW avoiding
>>> adjacent NW/SE and adjacent NE/SW
>>> Show that this satisfies the recurrence
>>> a(0)=1; for n>0: a(n) = 4*9^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).
>>>
>>> (The second and fourth questions are stated as if they are theorems, but
>>> no
>>> proof is given.)
>>>
>>> Neil
>>>
>>> --
>>> Seqfan Mailing list -
>>> https://nam11.safelinks.protection.outlook.com/?url=http%3A%2F%2Flist.seqfan.eu%2F&amp;data=01%7C01%7C%7Ce198304f85fa4fff06d108d804520b8a%7C74540637643546cc87a46d38efb78538%7C0&amp;sdata=DzmnJsOnapeI6w15s1%2FMrnFBFPcQvW%2FCkFOw6mLibDY%3D&amp;reserved=0
>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>


From pemd70 at yahoo.com  Sun Jun  7 21:52:17 2020
From: pemd70 at yahoo.com (Ali Sada)
Date: Sun, 7 Jun 2020 19:52:17 +0000 (UTC)
Subject: [seqfan] Connection between A000245 and A026016 (or A071725)?
References: <791425054.629861.1591559537899.ref@mail.yahoo.com>
Message-ID: <791425054.629861.1591559537899@mail.yahoo.com>

Hi Everyone,

Arrange the numbers between 1 and n in a raw of size 2n such that: 
a) 1 is always in the first spot;
b) k's position in the raw is <= 2k; and ?
c) k always comes before k+1


For example, below are the possibilities we get when n = 3. (I put ?*? instead of a space for more clarity.) 

1, 2, 3, *,* ,* 
1, 2, *, 3, *, *
1, 2, *, *, 3, * 
1, 2, *, *, *, 3
1, *, 2, 3, *, * 
1, *, 2, *, 3, *
1, *, 2, *, *, 3
1, *, *, 2, 3, * 
1, *, *, 2, *, 3

The possibilities I got were 1, 3, 9, 28, 90, 297, 1001, 3432, 11934 (A000245.)

Then, I filled the remaining spaces with the numbers between n+1 and 2n. 

Using n = 3 as an example we get: 

1, 2, 3, 4, 5, 6
1, 2, 4, 3, 5, 6
1, 2, 4, 5, 3, 6
1, 2, 4, 5, 6, 3
1, 4, 2, 3, 5, 6
1, 4, 2, 5, 3, 6
1, 4, 2, 5, 6, 3
1, 4, 5, 2, 3, 6
1, 4, 5, 2, 6, 3

For the numbers I tested (only 8,) the sum of the third column minus the sum of the second column was A026016(n-1).

In the example of n = 3 above, the sum of the third column is 31, and the sum of the second column is 28. 
31-28 = 3, which is A026016(2).

3 is also the result of the subtraction of the fourth column minus the third column, and the sixth column minus the fifth column. 

I would really appreciate it if you could tell me if this pattern will continue and why.

Best,

Ali





From baruchel at gmx.com  Mon Jun  8 11:07:34 2020
From: baruchel at gmx.com (Thomas Baruchel)
Date: Mon, 8 Jun 2020 11:07:34 +0200 (CEST)
Subject: [seqfan] Looking for a function expanding to (1-q^k)^(-1) terms
Message-ID: <alpine.DEB.2.11.2006081104480.374@raspberrypi>

Hi,

is someone aware of some function expanding to the following series:


     x/(1-q)  +  x^2 / (1-q^2)   +  x^3 / (1-q^3)   + ...

I have a close match with the QPolyGamma function which evaluates to the
sum above at x=1, but I am not sure I can fully build the given series from it.

Thank you by advance,

best regards,

--
Thomas Baruchel


From simon.plouffe at gmail.com  Mon Jun  8 11:34:54 2020
From: simon.plouffe at gmail.com (Simon Plouffe)
Date: Mon, 8 Jun 2020 11:34:54 +0200
Subject: [seqfan] Re: Looking for a function expanding to (1-q^k)^(-1) terms
In-Reply-To: <alpine.DEB.2.11.2006081104480.374@raspberrypi>
References: <alpine.DEB.2.11.2006081104480.374@raspberrypi>
Message-ID: <CAK+PtObiNVGWZSMdofAgek161SKU485p=oCBon6FANjSbrNvOg@mail.gmail.com>

hello Thomas,
but this is tau(n) * x^n, isn't ?
this is well known.
just expand the sum into a series. ...

Simon



Le lun. 8 juin 2020 ? 11:25, Thomas Baruchel <baruchel at gmx.com> a ?crit :

> Hi,
>
> is someone aware of some function expanding to the following series:
>
>
>      x/(1-q)  +  x^2 / (1-q^2)   +  x^3 / (1-q^3)   + ...
>
> I have a close match with the QPolyGamma function which evaluates to the
> sum above at x=1, but I am not sure I can fully build the given series
> from it.
>
> Thank you by advance,
>
> best regards,
>
> --
> Thomas Baruchel
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From techsubs at pearceneptune.co.uk  Mon Jun  8 18:01:07 2020
From: techsubs at pearceneptune.co.uk (Peter Munn)
Date: Mon, 8 Jun 2020 17:01:07 +0100
Subject: [seqfan] A107008 = primes of the form 24k+1?
Message-ID: <78b93da138ac4a07e4af5195cfb89d51.squirrel@webmail.nsnoc.com>

Hi seqfans.

In A107008 (Primes of the form x^2+24*y^2), NJAS comments "Presumably this
is the same as primes congruent to 1 mod 24."

Can we come up with something to decide this?

It would help to establish when 24k+1 can be written as x^2+24*y^2. I
reckon this happens when the k in 24k+1 can be written as the sum of a
square, j, and a generalized pentagonal number, i, because setting y =
sqrt(j), x = sqrt(24i+1) can be shown to satisfy the equation. (See Zak
Seidov's 2008 comment in http://oeis.org/A001318, "Generalized pentagonal
numbers".) I believe the converse is true, also.

OEIS does not yet have the sequence "Numbers that can't be written as the
sum of a square and a generalized pentagonal number" [1], but the
sequence, S, starts 20, 29, 33, 34, 45, 46, 53, ... , and I reckon it has
positive asymptotic density.

So the question becomes: if k is a term of S, why should 24k+1 be
composite, at least up to the limit of Vladimir Orlovsky's check in
A107008? From the first 3 terms we get: 20*24 + 1 = 481 = 13*37; 29*24 + 1
= 697 = 17*41; 33*24 + 1 = 793 = 13*61.

Going through more terms, I saw a pattern emerge, prompting me to ask: is
this particular subset of "24k+1" numbers the same as "nonsquare numbers
of the form (24i + m) * (24j + m), 0 <= i < j, m in {13, 17, 19, 23}"?
This would be interesting anyway, and could be a clue.

However, I'm not sure I'm close to an answer, and there might be a much
easier route: does anyone have better ideas? Or know the answer already?

Best regards,

Peter

[1] I also tried looking for the number of ways positive integers _can_ be
so written, using "Search: seq:2,2,1,1,2,2,1,1,2,1,2,1,1,1,1,4" but it
draws a blank.
























From techsubs at pearceneptune.co.uk  Mon Jun  8 20:30:14 2020
From: techsubs at pearceneptune.co.uk (Peter Munn)
Date: Mon, 8 Jun 2020 19:30:14 +0100
Subject: [seqfan] Re: A107008 = primes of the form 24k+1?
Message-ID: <05651d9a07c70d6a8df607d445185932.squirrel@webmail.nsnoc.com>

Following a private response on this subject, I should emphasize that the
tricky part seems to be to show that _all_ primes congruent to 1 mod 24
are in A107008. And after a little more spreadsheet work, it is starting
to look particularly interesting...

Does anyone want to check to what extent the following hypothesis is true?:

When k can be written as the sum of a square and a generalized pentagonal
number in exactly one way, the resulting numbers of the form 24k+1 might
be exactly the prime numbers of the form 24k+1 plus the squares of prime
numbers congruent to 13, 17, 19 or 23 mod 24.

Peter

On Mon, June 8, 2020 5:01 pm, Peter Munn wrote:
> Hi seqfans.
> In A107008 (Primes of the form x^2+24*y^2), NJAS comments "Presumably
this
> is the same as primes congruent to 1 mod 24."
> Can we come up with something to decide this?
> It would help to establish when 24k+1 can be written as x^2+24*y^2. I
reckon this happens when the k in 24k+1 can be written as the sum of a
square, j, and a generalized pentagonal number, i, because setting y =
sqrt(j), x = sqrt(24i+1) can be shown to satisfy the equation. (See Zak
Seidov's 2008 comment in http://oeis.org/A001318, "Generalized
pentagonal
> numbers".) I believe the converse is true, also.
> OEIS does not yet have the sequence "Numbers that can't be written as
the
> sum of a square and a generalized pentagonal number" [1], but the
sequence, S, starts 20, 29, 33, 34, 45, 46, 53, ... , and I reckon it
has
> positive asymptotic density.
> So the question becomes: if k is a term of S, why should 24k+1 be
composite, at least up to the limit of Vladimir Orlovsky's check in
A107008? From the first 3 terms we get: 20*24 + 1 = 481 = 13*37; 29*24 +
1
> = 697 = 17*41; 33*24 + 1 = 793 = 13*61.
> Going through more terms, I saw a pattern emerge, prompting me to ask:
is
> this particular subset of "24k+1" numbers the same as "nonsquare numbers
of the form (24i + m) * (24j + m), 0 <= i < j, m in {13, 17, 19, 23}"?
This would be interesting anyway, and could be a clue.
> However, I'm not sure I'm close to an answer, and there might be a much
easier route: does anyone have better ideas? Or know the answer already?
Best regards,
> Peter
> [1] I also tried looking for the number of ways positive integers _can_
be
> so written, using "Search: seq:2,2,1,1,2,2,1,1,2,1,2,1,1,1,1,4" but it
draws a blank.
> --
> Seqfan Mailing list - http://list.seqfan.eu/






From wouter.meeussen at telenet.be  Tue Jun  9 12:11:42 2020
From: wouter.meeussen at telenet.be (Wouter Meeussen)
Date: Tue, 9 Jun 2020 12:11:42 +0200
Subject: [seqfan] Re: interesting idea, weird ramifications, as yet unsolved
In-Reply-To: <1457329491.2128123.1591680009085@mail1.libero.it>
References: <D9FBE905A7AB491997B0748B761F565B@Wou>
 <1457329491.2128123.1591680009085@mail1.libero.it>
Message-ID: <A85A70B168864715B67DF39D26B97C60@Wou>

We can look at the question asked by 'Cesare' in a wider scope:

define xx(0) as the vector {x_1, ... , x_n} and define xx(1) as the outer 
product Sum(i=1..n,j=1..n;  x_i x_j ) , subtract the diagonal elements 
Sum(i=1..n ; x_i x_i) and divide by two. This extracts only the terms below 
the diagonal of the outer product.
Repeat this procedure up to xx(k).
The result is a symmetric polynomial in n variables of degree 2^k.

The number of terms (monomials) as function of n and k is :

n=3 ; 3,  3,   3,      3, ...
n=4 ; 4,  6, 15,  102,      5010,          12 367 734, 76 069 184 710 488, 
...
n=5 ; 5, 10, 45, 975, 470 025, 110 131 246 500, ...


for n=4, there are 12 monomials containing 3 variables, all others have 4 
variables.
for n=5, I find 30 monomials in 3 variables, rest in 4 or 5 variables
(all for k>1 of course)

nice detail:
the monomials in 3 variables are simple when written in terms of the 
monomial symmetric functions:
for all k and n=2,...,7 they are m({1,1}) , m({2,1,1}), m({3,3,2}), 
m({6,5,5}), m({11,11,10}), m({22,21,21}) etc.

Wouter.
(sanity checks are always appreciated)
__________________________________
in Mathematica:
Remove[X];n=4;
X[0] = Array[Subscript[x, #] &, n];
X[k_Integer /; k > 0] :=  X[k] =
List @@ Expand[(Apply[Plus, Flatten[Outer[Times, X[k - 1], X[k - 1]]]] - 
Dot[X[k - 1], X[k - 1]])/2 ];

__________________________________

-----Original Message----- 
From: Luca Petrone
Sent: Tuesday, June 09, 2020 7:20 AM
To: Sequence Fanatics Discussion list ; Wouter Meeussen
Subject: Re: [seqfan] interesting idea, weird ramifications, as yet unsolved

Of course there are "easy" formulas to calculate how many elements of X_N 
have, for example, k repetition of one element, but things get non-sensely 
complicated of you are asking k repetions of one item, l repetitions of 
another item and so on: is really this the question? and what is it the 
meaning of it?
Best Regards,
Luca
> Il 7 giugno 2020 alle 20.16 Wouter Meeussen <wouter.meeussen at telenet.be> 
> ha scritto:
>
>
> ?Pairwise combinations of distinct elements?
> see https://math.stackexchange.com/questions/3708151
>
> my feeling is that Algebraic Combinatorics should clarify this, reducing 
> it to a well known pattern.
> I tried but failed dismally.
>
> It boils down to a list of monomials X_(k+1) generated by
>
> (X_k ** X_k  - X_k . X_k )/2 where ?**? represents the outer product.
>
> The author (?Cesare?) only asks for a very limited property of these 
> multinomials, but there is a lot of other ?enumerative combinatorics? 
> going on here.
>
> Does anyone recognise the patten?
>
> Wouter.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/ 



From pemd70 at yahoo.com  Wed Jun 10 23:07:49 2020
From: pemd70 at yahoo.com (Ali Sada)
Date: Wed, 10 Jun 2020 21:07:49 +0000 (UTC)
Subject: [seqfan] All-Fives Domino Sequence
References: <1975428622.2422057.1591823269175.ref@mail.yahoo.com>
Message-ID: <1975428622.2422057.1591823269175@mail.yahoo.com>

Hi Everyone,

In Iraq, we play a variation of the All-Five game called Aznif. (Some scholars say the name comes of an Armenian word that means ?noble lady?). 

These are the rules:

1)?? ?You can start with any tile.
2)?? ?The first double on the ground becomes the (only) spinner.
3)?? ?The goal is to score as many multiples of 5 by adding the numbers on the ends in each step. (The points on the spinner don?t count after it is covered from 2 sides).


In this variation, we can score 470 points using all 28 tiles of the order 6 domino set. These pictures show how, step by step: 

https://justpaste.it/2ipxj

I don't have a proof, but I think we can?t score more than 470.

The sequence here is the total points we can score using a domino set of order n.

For, n = 0, we have only one tile 0?0, and we can?t score anything. So, a(0) = 0.
For, n = 1, we have three tiles, and we can?t score anything. So, a(1) = 0.
For n = 2, we have six tiles. We can only score 5 once. So, a(2) = 5.
For n = 3, we have ten tiles. We can score, 5+5+10+10+10+10. So, a(3) = 50.

And so on.
The question is: can we find a way to count the scores for domino sets with higher orders (other than trial and error)?
Also, is this sequence suitable for the OEIS?

Best,

Ali




From luca.petrone at libero.it  Tue Jun  9 07:20:09 2020
From: luca.petrone at libero.it (Luca Petrone)
Date: Tue, 9 Jun 2020 07:20:09 +0200 (CEST)
Subject: [seqfan] Re: interesting idea, weird ramifications, as yet unsolved
In-Reply-To: <D9FBE905A7AB491997B0748B761F565B@Wou>
References: <D9FBE905A7AB491997B0748B761F565B@Wou>
Message-ID: <1457329491.2128123.1591680009085@mail1.libero.it>

Of course there are "easy" formulas to calculate how many elements of X_N have, for example, k repetition of one element, but things get non-sensely complicated of you are asking k repetions of one item, l repetitions of another item and so on: is really this the question? and what is it the meaning of it?
Best Regards,
Luca
> Il 7 giugno 2020 alle 20.16 Wouter Meeussen <wouter.meeussen at telenet.be> ha scritto:
> 
> 
> ?Pairwise combinations of distinct elements?
> see https://math.stackexchange.com/questions/3708151
> 
> my feeling is that Algebraic Combinatorics should clarify this, reducing it to a well known pattern.
> I tried but failed dismally.
> 
> It boils down to a list of monomials X_(k+1) generated by
> 
> (X_k ** X_k  - X_k . X_k )/2 where ?**? represents the outer product.
> 
> The author (?Cesare?) only asks for a very limited property of these multinomials, but there is a lot of other ?enumerative combinatorics? going on here.
> 
> Does anyone recognise the patten?
> 
> Wouter.
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/


From bradklee at gmail.com  Wed Jun 17 04:43:26 2020
From: bradklee at gmail.com (Brad Klee)
Date: Tue, 16 Jun 2020 21:43:26 -0500
Subject: [seqfan] Ramanujan-Inspired Sequences for 1/pi.
Message-ID: <CAC_LFVasF-og0ChFyh4XNf_DRufXBkYDsTx+uAx1PyaeVpm+Uw@mail.gmail.com>

s=2: 1, 6, 56, 620, 7512, 96208, 1279168, 17471448, 243509720, 3447792656,
49434765888 . . .   nAn.

s=3: 1, 9, 138, 2550, 51840, 1116612, 24999408, 575368596, 13518747000,
322765065480, 7805239515216 . . .   nAn.

s=4: 1, 18, 632, 27300, 1306200, 66413424, 3515236032, 191434588488,
10648603594200, 602109586993200, 34491343330027584 . . .   nAn.

s=6: 1, 90, 20280, 5798100, 1854085464, 632693421360, 225235329359040,
82598530506097320, 30962429500615006680, 11803615010304909757680,
4560219108801622243897920 . . .   nAn.

They are all double binomial sums, but can be reduced to single sums using
Zeilberger's algorithm. Dividing a(n) by c^n for c=32,54,128, or 864,
summing
over n, we obtain either 4/pi, 9*sqrt(3)/(4*pi), 8*sqrt(2)/(3*pi) or
18/(5*pi).

These are a fairly straightforward consequence of the assertion in S.14 of:
http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf

More on this soon...

--Brad


From bpcrtz at free.fr  Mon Jun  1 14:14:56 2020
From: bpcrtz at free.fr (Paul Curtz)
Date: Mon, 01 Jun 2020 12:14:56 -0000
Subject: [seqfan] Pseudotriangle
In-Reply-To: <1647911946.153700598.1591006184939.JavaMail.root@spooler7-g27.priv.proxad.net>
Message-ID: <1834687573.153960332.1591013696293.JavaMail.root@spooler7-g27.priv.proxad.net>

Hello,
                                   sum
   0                                0
   0                                0
   0  1                             1
   0  2  2                          4
   0  3  4                          7
   0  4  6  3                      13
   0  5  8  6  4                   23
   0  6 10  9  8                   33
   0  7 12 12 12  5                48
   0  8 14 15 16 10  6             69
   0  9 16 18 20 15 12             90
   ...
  b(n) = 0, 0, A038391(n).

b(n) first two differences:
  0,  0,  1,  4,  7, 13, 23, 33, 48, 69, 90, 118, 154, ...
  0,  1,  3,  3,  6, 10, 10, 15, 21, 21, 28,  36,  36, ... (from A000217(n))
  1,  2,  0,  3,  4,  0,  5,  6,  0,  7,  8,   0,   9, ... = A117909(n).

Is this kind of partial triangle known?

Best regards

Paul


From pemd70 at yahoo.com  Fri Jun 19 04:10:01 2020
From: pemd70 at yahoo.com (Ali Sada)
Date: Fri, 19 Jun 2020 02:10:01 +0000 (UTC)
Subject: [seqfan] Four grid-based sequences
References: <401142798.530443.1592532601323.ref@mail.yahoo.com>
Message-ID: <401142798.530443.1592532601323@mail.yahoo.com>

Hi Everyone,

Please see the 4 sequences below. I need help with their definitions and with finding more terms. Sorry in advance if the terminology I am using is not proper.? 

1.?? ?In the corner of a 2D grid, we put 1. Now, we put the other numbers such that n is connected to n-1. Symmetry is not allowed.
 We start with a(1) = 1. 

Now, 2 has 2 potential squares: (1,2) and (2,2). So, a(2) = 2.
Based on the places of 2, 3 will have 8 potential squares.
4 will have 38 potential squares, and 5 will have 196, and so on.
1 ,2 , 8, 38, 196, 

These shapes might explain the idea in a better way: 
https://justpaste.it/6qxc0


2.?? ?The same idea of the first sequence except that n could be connected to any number already on the grid. Symmetry is not allowed.
The sequence here will be 1, 2, 8, 53, 463, 

These are the shapes generated by this algorithm ?
https://justpaste.it/6qju4

3.?? ?The same idea of the first sequence but the grid here has no edges. We put 1 in any square on the grid and we continue. n should be connected to n-1, and symmetry is not allowed. 

The sequence here is
1, 2, 8, 47, 295,? 

These are the shapes generated by this algorithm ?
https://justpaste.it/4l3p4


4.?? ?In this version, the grid here also has no edges, n could be connected to any number already on the grid, and symmetry is not allowed. 

The sequence here is 
1, 2, 13, 154, 2419,
These are the shapes generated by this algorithm 
https://justpaste.it/2b92w
(I know it seems like self-serving, but I really like the shapes of this version!)

I found the terms of the sequences above ?by hand?, so the possibility of a mistake is there, as usual.

I also need help with two more sequences. The same ideas of sequences 2 and 4, but with one symbol (e.g. ?*?) instead of numbers. 

Best,

Ali





















From jean-luc.manguin at unicaen.fr  Fri Jun 19 09:37:11 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Fri, 19 Jun 2020 09:37:11 +0200 (CEST)
Subject: [seqfan] Re: Four grid-based sequences
In-Reply-To: <401142798.530443.1592532601323@mail.yahoo.com>
References: <401142798.530443.1592532601323.ref@mail.yahoo.com>
 <401142798.530443.1592532601323@mail.yahoo.com>
Message-ID: <368101878.7333177.1592552231063.JavaMail.zimbra@unicaen.fr>

Hello Ali,

Your problem is (IMO) a variation of the problem of polyominoes ; the differences are :
- the nodes are labelled with numbers.
- corner connections are allowed

So the model to use in this case is graph-based, with labelled nodes.

The labels make things a bit more complicated, but as you also consider symmetry or not, it can be solved by a backtracking algorithm (transfer matrix algo is much faster but symmetry is difficult to take in account, maybe impossible...)
If I had to solve it, I would consider the problem without labels and make an implementation of the algorithm. After that you could add the constraints with the labels.

Best regards,

JLM

----- Mail original -----
De: "seqfan" <seqfan at list.seqfan.eu>
?: "seqfan" <seqfan at list.seqfan.eu>
Cc: "Ali Sada" <pemd70 at yahoo.com>
Envoy?: Vendredi 19 Juin 2020 04:10:01
Objet: [seqfan] Four grid-based sequences

Hi Everyone,

Please see the 4 sequences below. I need help with their definitions and with finding more terms. Sorry in advance if the terminology I am using is not proper.? 

1.?? ?In the corner of a 2D grid, we put 1. Now, we put the other numbers such that n is connected to n-1. Symmetry is not allowed.
 We start with a(1) = 1. 

Now, 2 has 2 potential squares: (1,2) and (2,2). So, a(2) = 2.
Based on the places of 2, 3 will have 8 potential squares.
4 will have 38 potential squares, and 5 will have 196, and so on.
1 ,2 , 8, 38, 196, 

These shapes might explain the idea in a better way: 
https://justpaste.it/6qxc0


2.?? ?The same idea of the first sequence except that n could be connected to any number already on the grid. Symmetry is not allowed.
The sequence here will be 1, 2, 8, 53, 463, 

These are the shapes generated by this algorithm ?
https://justpaste.it/6qju4

3.?? ?The same idea of the first sequence but the grid here has no edges. We put 1 in any square on the grid and we continue. n should be connected to n-1, and symmetry is not allowed. 

The sequence here is
1, 2, 8, 47, 295,? 

These are the shapes generated by this algorithm ?
https://justpaste.it/4l3p4


4.?? ?In this version, the grid here also has no edges, n could be connected to any number already on the grid, and symmetry is not allowed. 

The sequence here is 
1, 2, 13, 154, 2419,
These are the shapes generated by this algorithm 
https://justpaste.it/2b92w
(I know it seems like self-serving, but I really like the shapes of this version!)

I found the terms of the sequences above ?by hand?, so the possibility of a mistake is there, as usual.

I also need help with two more sequences. The same ideas of sequences 2 and 4, but with one symbol (e.g. ?*?) instead of numbers. 

Best,

Ali




















--
Seqfan Mailing list - http://list.seqfan.eu/




From sairvin at gmail.com  Mon Jun 22 03:51:20 2020
From: sairvin at gmail.com (Sean A. Irvine)
Date: Mon, 22 Jun 2020 13:51:20 +1200
Subject: [seqfan] A032435 and A032436 Josephus problem data mismatch
Message-ID: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>

Hi all,

The triangles in A032434 through A032436 concern the Josephus problem.
That's the problem where n people stand in a circle and then you go round
and round killing every k-th person until nearly everyone is dead.

The data in A032435 and A032436 do not match what I expect.

There are links from these sequences into Mathworld.  The Mathworld entry
contains two triangles which do contain the data I expect and which also
point back to these OEIS entries.  Neither of the triangles in the
Mathworld entry actually seem to be present in the OEIS.

However, we don't simply want to replace the existing data in A032435 and
A032436, since the current data values are likely related to this problem
somehow.  I spent a couple of hours looking at the current data values and
could not determine an explanation, perhaps someone else will have more
success.

Sean.


From njasloane at gmail.com  Mon Jun 22 04:52:31 2020
From: njasloane at gmail.com (Neil Sloane)
Date: Sun, 21 Jun 2020 22:52:31 -0400
Subject: [seqfan] Re: A032435 and A032436 Josephus problem data mismatch
In-Reply-To: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>
References: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>
Message-ID: <CAAOnSgTJcoe2xiffY-s4J-=h_Q9RDNLszLOz30Wc67ANeVAdew@mail.gmail.com>

The refs for A032435 are Ball-Coxeter, and one of Kraitchik's books.  I
checked Ball-Coxeter, but it was not helpful (though I only skimmed it).
I have three of Kraitchik's books, but apparently not that one.  So that
would be the next place to check.  Maybe it is online somewhere?

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sun, Jun 21, 2020 at 9:51 PM Sean A. Irvine <sairvin at gmail.com> wrote:

> Hi all,
>
> The triangles in A032434 through A032436 concern the Josephus problem.
> That's the problem where n people stand in a circle and then you go round
> and round killing every k-th person until nearly everyone is dead.
>
> The data in A032435 and A032436 do not match what I expect.
>
> There are links from these sequences into Mathworld.  The Mathworld entry
> contains two triangles which do contain the data I expect and which also
> point back to these OEIS entries.  Neither of the triangles in the
> Mathworld entry actually seem to be present in the OEIS.
>
> However, we don't simply want to replace the existing data in A032435 and
> A032436, since the current data values are likely related to this problem
> somehow.  I spent a couple of hours looking at the current data values and
> could not determine an explanation, perhaps someone else will have more
> success.
>
> Sean.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From amiram.eldar at gmail.com  Mon Jun 22 07:18:47 2020
From: amiram.eldar at gmail.com (Ami Eldar)
Date: Mon, 22 Jun 2020 08:18:47 +0300
Subject: [seqfan] Re: A032435 and A032436 Josephus problem data mismatch
In-Reply-To: <CAAOnSgTJcoe2xiffY-s4J-=h_Q9RDNLszLOz30Wc67ANeVAdew@mail.gmail.com>
References: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>
 <CAAOnSgTJcoe2xiffY-s4J-=h_Q9RDNLszLOz30Wc67ANeVAdew@mail.gmail.com>
Message-ID: <CAMJ5+vfhTtRSCoW1BM1zJiDLJ6+-CYyt=MNDBhsu-wYN5YuWQQ@mail.gmail.com>

The book "Mathematical Recreations" by  Kraitchik can be borrowed (freely
browsed for a limited time) at the Internet Archive:
https://archive.org/details/mathematicalrecr0002krai
https://archive.org/details/mathematicalrecr0000krai
https://archive.org/details/mathematicalrecr00krai



On Mon, Jun 22, 2020 at 5:52 AM Neil Sloane <njasloane at gmail.com> wrote:

> The refs for A032435 are Ball-Coxeter, and one of Kraitchik's books.  I
> checked Ball-Coxeter, but it was not helpful (though I only skimmed it).
> I have three of Kraitchik's books, but apparently not that one.  So that
> would be the next place to check.  Maybe it is online somewhere?
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Sun, Jun 21, 2020 at 9:51 PM Sean A. Irvine <sairvin at gmail.com> wrote:
>
> > Hi all,
> >
> > The triangles in A032434 through A032436 concern the Josephus problem.
> > That's the problem where n people stand in a circle and then you go round
> > and round killing every k-th person until nearly everyone is dead.
> >
> > The data in A032435 and A032436 do not match what I expect.
> >
> > There are links from these sequences into Mathworld.  The Mathworld entry
> > contains two triangles which do contain the data I expect and which also
> > point back to these OEIS entries.  Neither of the triangles in the
> > Mathworld entry actually seem to be present in the OEIS.
> >
> > However, we don't simply want to replace the existing data in A032435 and
> > A032436, since the current data values are likely related to this problem
> > somehow.  I spent a couple of hours looking at the current data values
> and
> > could not determine an explanation, perhaps someone else will have more
> > success.
> >
> > Sean.
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From olivier.gerard at gmail.com  Mon Jun 22 08:24:10 2020
From: olivier.gerard at gmail.com (Olivier Gerard)
Date: Mon, 22 Jun 2020 09:24:10 +0300
Subject: [seqfan] Re: A032435 and A032436 Josephus problem data mismatch
In-Reply-To: <CAMJ5+vfhTtRSCoW1BM1zJiDLJ6+-CYyt=MNDBhsu-wYN5YuWQQ@mail.gmail.com>
References: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>
 <CAAOnSgTJcoe2xiffY-s4J-=h_Q9RDNLszLOz30Wc67ANeVAdew@mail.gmail.com>
 <CAMJ5+vfhTtRSCoW1BM1zJiDLJ6+-CYyt=MNDBhsu-wYN5YuWQQ@mail.gmail.com>
Message-ID: <CAAcpPSbmtHq62EBYipF1NiOpE6V=L_3FmjAOxhyC4kUP+xCJww@mail.gmail.com>

Thanks to Jean-Paul Allouche who found it by other means, I have
now forwarded the relevant page to all interested parties.

If there are other people wanting to read it, I will forward it to them.

Olivier G?rard
Seqfan Mailing List Administrator



On Mon, Jun 22, 2020 at 9:00 AM Ami Eldar <amiram.eldar at gmail.com> wrote:

> The book "Mathematical Recreations" by  Kraitchik can be borrowed (freely
> browsed for a limited time) at the Internet Archive:
> https://archive.org/details/mathematicalrecr0002krai
> https://archive.org/details/mathematicalrecr0000krai
> https://archive.org/details/mathematicalrecr00krai
>
>
>
> On Mon, Jun 22, 2020 at 5:52 AM Neil Sloane <njasloane at gmail.com> wrote:
>
> > The refs for A032435 are Ball-Coxeter, and one of Kraitchik's books.  I
> > checked Ball-Coxeter, but it was not helpful (though I only skimmed it).
> > I have three of Kraitchik's books, but apparently not that one.  So that
> > would be the next place to check.  Maybe it is online somewhere?
> >
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, President, OEIS Foundation.
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > Email: njasloane at gmail.com
> >
> >
> >
> > On Sun, Jun 21, 2020 at 9:51 PM Sean A. Irvine <sairvin at gmail.com>
> wrote:
> >
> > > Hi all,
> > >
> > > The triangles in A032434 through A032436 concern the Josephus problem.
> > > That's the problem where n people stand in a circle and then you go
> round
> > > and round killing every k-th person until nearly everyone is dead.
> > >
> > > The data in A032435 and A032436 do not match what I expect.
> > >
> > > There are links from these sequences into Mathworld.  The Mathworld
> entry
> > > contains two triangles which do contain the data I expect and which
> also
> > > point back to these OEIS entries.  Neither of the triangles in the
> > > Mathworld entry actually seem to be present in the OEIS.
> > >
> > > However, we don't simply want to replace the existing data in A032435
> and
> > > A032436, since the current data values are likely related to this
> problem
> > > somehow.  I spent a couple of hours looking at the current data values
> > and
> > > could not determine an explanation, perhaps someone else will have more
> > > success.
> > >
> > > Sean.
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From DavidRabahy at comcast.net  Tue Jun 23 20:40:13 2020
From: DavidRabahy at comcast.net (David Rabahy)
Date: Tue, 23 Jun 2020 14:40:13 -0400
Subject: [seqfan] like A004080 but for 1/(2i) and more
Message-ID: <001901d6498d$bbad4700$3307d500$@comcast.net>

Least k such that H2(k) >= n, where H2(k) is the even harmonic number sum_{i=1..k} 1/(2i);
n
1 - 5
2 - 31
3 - 227
4 - 1674
...

Is this worthy of an entry in the OEIS or too trivial?

https://docs.google.com/spreadsheets/d/1GbRABHLdAtFHU0dFDkHqanJYje15L3gbc6RrWzWCnXo

Clearly this idea could be generalized, i.e. where Hq(k) is the qth harmonic number sum_{i=1..k} 1/(qi).

Also, it could be diagonalized;
n
1 - 1
2 - 31
3 - 4550
...



From seqfan at hasler.fr  Wed Jun 24 05:47:41 2020
From: seqfan at hasler.fr (M. F. Hasler)
Date: Tue, 23 Jun 2020 23:47:41 -0400
Subject: [seqfan] Re: like A004080 but for 1/(2i) and more
In-Reply-To: <001901d6498d$bbad4700$3307d500$@comcast.net>
References: <001901d6498d$bbad4700$3307d500$@comcast.net>
Message-ID: <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>

On Tue, Jun 23, 2020 at 10:33 PM David Rabahy <DavidRabahy at comcast.net>
wrote:

> Least k such that H2(k) >= n, where H2(k) is the even harmonic number
> sum_{i=1..k} 1/(2i);
> n
> 1 - 5
> 2 - 31
> 3 - 227
> 4 - 1674
>

Unless I'm wrong it should be 4, not 5, for n=1 above,
and more generally, A4080(2n), for obvious reasons :

Clearly this idea could be generalized, i.e. where Hq(k) is the qth
> harmonic number sum_{i=1..k} 1/(qi).
>

which is the same as H(k)/q, whence your sequence should be A4080(qn).

- Maximilian


From jean-luc.manguin at unicaen.fr  Wed Jun 24 15:20:03 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Wed, 24 Jun 2020 15:20:03 +0200 (CEST)
Subject: [seqfan] Polyominoes (A000105, A000988, A001168) - offset problem
Message-ID: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>

Hello, 

Everytime I edit a sequence, I receive justified remarks about my non-respect of some rules, although of course this non-respect is not done by purpose. 

In return, I would like to find in OEIS a care about coherence ; this example needs (IMO) a little correction : 
- A000105 has an offset = 0 
- A000988 and A001168 have offset = 1 

These three sequences give the number of polyominoes (free, one-sided, and fixed) and as it is written in the dedicated page in Mathworld, [ https://mathworld.wolfram.com/Polyomino.html | https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino with zero cell ; then I would be pleased to see A000105 with offset = 1, which seems much more logical. Of course I could edit this sequence (I did not try, I guess I can), but I prefer to mention this in the list. 

Thank you, dear editors, for your attention to this message. 
Have a nice day and a happy St John ! 

JL Manguin 



From fred.lunnon at gmail.com  Wed Jun 24 17:51:37 2020
From: fred.lunnon at gmail.com (Fred Lunnon)
Date: Wed, 24 Jun 2020 16:51:37 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
Message-ID: <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>

  A000105 with offset = 0  is correct as it currently stands:
there is 1 empty polyomino with  #cells = 0 .

  My preference would be to attach an extra 1 at the front of both
A000988 and A001168 , and edit to  offset = 0.

  All three should cross-refer to the others; and similar remarks
may apply to related sequences, such as A030228.   NJAS ?

Fred Lunnon


On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> Hello,
>
> Everytime I edit a sequence, I receive justified remarks about my
> non-respect of some rules, although of course this non-respect is not done
> by purpose.
>
> In return, I would like to find in OEIS a care about coherence ; this
> example needs (IMO) a little correction :
> - A000105 has an offset = 0
> - A000988 and A001168 have offset = 1
>
> These three sequences give the number of polyominoes (free, one-sided, and
> fixed) and as it is written in the dedicated page in Mathworld, [
> https://mathworld.wolfram.com/Polyomino.html |
> https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino with
> zero cell ; then I would be pleased to see A000105 with offset = 1, which
> seems much more logical. Of course I could edit this sequence (I did not
> try, I guess I can), but I prefer to mention this in the list.
>
> Thank you, dear editors, for your attention to this message.
> Have a nice day and a happy St John !
>
> JL Manguin
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From sairvin at gmail.com  Wed Jun 24 21:00:50 2020
From: sairvin at gmail.com (Sean A. Irvine)
Date: Thu, 25 Jun 2020 07:00:50 +1200
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
Message-ID: <CAL0FETkCXE48Aqsv-2_hcXds3NbA6UdB3Peh+Xvy8Tm=o14cAA@mail.gmail.com>

I'm with Fred on this.  The current values are all correct and consistent,
but indeed a(0)=1 could be inserted as Fred suggests.


On Thu, 25 Jun 2020 at 03:51, Fred Lunnon <fred.lunnon at gmail.com> wrote:

>   A000105 with offset = 0  is correct as it currently stands:
> there is 1 empty polyomino with  #cells = 0 .
>
>   My preference would be to attach an extra 1 at the front of both
> A000988 and A001168 , and edit to  offset = 0.
>
>   All three should cross-refer to the others; and similar remarks
> may apply to related sequences, such as A030228.   NJAS ?
>
> Fred Lunnon
>
>
> On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> > Hello,
> >
> > Everytime I edit a sequence, I receive justified remarks about my
> > non-respect of some rules, although of course this non-respect is not
> done
> > by purpose.
> >
> > In return, I would like to find in OEIS a care about coherence ; this
> > example needs (IMO) a little correction :
> > - A000105 has an offset = 0
> > - A000988 and A001168 have offset = 1
> >
> > These three sequences give the number of polyominoes (free, one-sided,
> and
> > fixed) and as it is written in the dedicated page in Mathworld, [
> > https://mathworld.wolfram.com/Polyomino.html |
> > https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino
> with
> > zero cell ; then I would be pleased to see A000105 with offset = 1, which
> > seems much more logical. Of course I could edit this sequence (I did not
> > try, I guess I can), but I prefer to mention this in the list.
> >
> > Thank you, dear editors, for your attention to this message.
> > Have a nice day and a happy St John !
> >
> > JL Manguin
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From rgwv at rgwv.com  Wed Jun 24 21:22:59 2020
From: rgwv at rgwv.com (rgwv at rgwv.com)
Date: Wed, 24 Jun 2020 15:22:59 -0400
Subject: [seqfan] Re: like A004080 but for 1/(2i) and more
In-Reply-To: <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>
References: <001901d6498d$bbad4700$3307d500$@comcast.net>
 <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>
Message-ID: <242501d64a5c$e0b763d0$a2262b70$@rgwv.com>

This is just the bisection of A002387.

-----Original Message-----
From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of M. F. Hasler
Sent: Tuesday, June 23, 2020 11:48 PM
To: DavidRabahy at comcast.net
Cc: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: like A004080 but for 1/(2i) and more

On Tue, Jun 23, 2020 at 10:33 PM David Rabahy <DavidRabahy at comcast.net>
wrote:

> Least k such that H2(k) >= n, where H2(k) is the even harmonic number 
> sum_{i=1..k} 1/(2i); n
> 1 - 5
> 2 - 31
> 3 - 227
> 4 - 1674
>

Unless I'm wrong it should be 4, not 5, for n=1 above,
and more generally, A4080(2n), for obvious reasons :

Clearly this idea could be generalized, i.e. where Hq(k) is the qth
> harmonic number sum_{i=1..k} 1/(qi).
>

which is the same as H(k)/q, whence your sequence should be A4080(qn).

- Maximilian

--
Seqfan Mailing list - http://list.seqfan.eu/



From njasloane at gmail.com  Wed Jun 24 21:47:55 2020
From: njasloane at gmail.com (Neil Sloane)
Date: Wed, 24 Jun 2020 15:47:55 -0400
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAL0FETkCXE48Aqsv-2_hcXds3NbA6UdB3Peh+Xvy8Tm=o14cAA@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <CAL0FETkCXE48Aqsv-2_hcXds3NbA6UdB3Peh+Xvy8Tm=o14cAA@mail.gmail.com>
Message-ID: <CAAOnSgQbwCbG7Ztd1sr38tp7eK=Pgu8NgMdKmXcBfwHZ99WxTg@mail.gmail.com>

I will follow Fred's advice, as always.


Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Wed, Jun 24, 2020 at 3:01 PM Sean A. Irvine <sairvin at gmail.com> wrote:

> I'm with Fred on this.  The current values are all correct and consistent,
> but indeed a(0)=1 could be inserted as Fred suggests.
>
>
> On Thu, 25 Jun 2020 at 03:51, Fred Lunnon <fred.lunnon at gmail.com> wrote:
>
> >   A000105 with offset = 0  is correct as it currently stands:
> > there is 1 empty polyomino with  #cells = 0 .
> >
> >   My preference would be to attach an extra 1 at the front of both
> > A000988 and A001168 , and edit to  offset = 0.
> >
> >   All three should cross-refer to the others; and similar remarks
> > may apply to related sequences, such as A030228.   NJAS ?
> >
> > Fred Lunnon
> >
> >
> > On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> > > Hello,
> > >
> > > Everytime I edit a sequence, I receive justified remarks about my
> > > non-respect of some rules, although of course this non-respect is not
> > done
> > > by purpose.
> > >
> > > In return, I would like to find in OEIS a care about coherence ; this
> > > example needs (IMO) a little correction :
> > > - A000105 has an offset = 0
> > > - A000988 and A001168 have offset = 1
> > >
> > > These three sequences give the number of polyominoes (free, one-sided,
> > and
> > > fixed) and as it is written in the dedicated page in Mathworld, [
> > > https://mathworld.wolfram.com/Polyomino.html |
> > > https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino
> > with
> > > zero cell ; then I would be pleased to see A000105 with offset = 1,
> which
> > > seems much more logical. Of course I could edit this sequence (I did
> not
> > > try, I guess I can), but I prefer to mention this in the list.
> > >
> > > Thank you, dear editors, for your attention to this message.
> > > Have a nice day and a happy St John !
> > >
> > > JL Manguin
> > >
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From jean-luc.manguin at unicaen.fr  Wed Jun 24 17:59:13 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Wed, 24 Jun 2020 17:59:13 +0200 (CEST)
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
Message-ID: <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>

Again, I refer to the definition of polyominoes (https://en.wikipedia.org/wiki/Polyomino) : "A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge."

"One or more", this is clear... (unless someone can prove that zero is more than one)

I agree that they should cross-refer (and that is another reason to put the same offset value).

JLM

----- Mail original -----
De: "Fred Lunnon" <fred.lunnon at gmail.com>
?: "seqfan" <seqfan at list.seqfan.eu>
Envoy?: Mercredi 24 Juin 2020 17:51:37
Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset problem

A000105 with offset = 0  is correct as it currently stands:
there is 1 empty polyomino with  #cells = 0 .

  My preference would be to attach an extra 1 at the front of both
A000988 and A001168 , and edit to  offset = 0.

  All three should cross-refer to the others; and similar remarks
may apply to related sequences, such as A030228.   NJAS ?

Fred Lunnon


On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> Hello,
>
> Everytime I edit a sequence, I receive justified remarks about my
> non-respect of some rules, although of course this non-respect is not done
> by purpose.
>
> In return, I would like to find in OEIS a care about coherence ; this
> example needs (IMO) a little correction :
> - A000105 has an offset = 0
> - A000988 and A001168 have offset = 1
>
> These three sequences give the number of polyominoes (free, one-sided, and
> fixed) and as it is written in the dedicated page in Mathworld, [
> https://mathworld.wolfram.com/Polyomino.html |
> https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino with
> zero cell ; then I would be pleased to see A000105 with offset = 1, which
> seems much more logical. Of course I could edit this sequence (I did not
> try, I guess I can), but I prefer to mention this in the list.
>
> Thank you, dear editors, for your attention to this message.
> Have a nice day and a happy St John !
>
> JL Manguin
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Seqfan Mailing list - http://list.seqfan.eu/




From davidrabahy at comcast.net  Wed Jun 24 19:25:36 2020
From: davidrabahy at comcast.net (davidrabahy at comcast.net)
Date: Wed, 24 Jun 2020 13:25:36 -0400
Subject: [seqfan] Re: like A004080 but for 1/(2i) and more
In-Reply-To: <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>
References: <001901d6498d$bbad4700$3307d500$@comcast.net>
 <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>
Message-ID: <005801d64a4c$7d6eed80$784cc880$@comcast.net>

Of course you are correct; I?m not sure how I got myself confused.  A014537 is the sequence I was looking for.

 

Is the diagonalization 1,3,4550,? interesting enough for an OEIS entry?

 

From: M. F. Hasler <seqfan at hasler.fr> 
Sent: Tuesday, June 23, 2020 11:48 PM
To: DavidRabahy at comcast.net
Cc: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: Re: [seqfan] like A004080 but for 1/(2i) and more

 

On Tue, Jun 23, 2020 at 10:33 PM David Rabahy <DavidRabahy at comcast.net <mailto:DavidRabahy at comcast.net> > wrote:

Least k such that H2(k) >= n, where H2(k) is the even harmonic number sum_{i=1..k} 1/(2i);
n
1 - 5
2 - 31
3 - 227
4 - 1674

 

Unless I'm wrong it should be 4, not 5, for n=1 above,

and more generally, A4080(2n), for obvious reasons :

 

Clearly this idea could be generalized, i.e. where Hq(k) is the qth harmonic number sum_{i=1..k} 1/(qi).

 

which is the same as H(k)/q, whence your sequence should be A4080(qn).

 

- Maximilian

 



From njasloane at gmail.com  Thu Jun 25 09:27:00 2020
From: njasloane at gmail.com (Neil Sloane)
Date: Thu, 25 Jun 2020 03:27:00 -0400
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
Message-ID: <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>

Everything in the Wikipedia is inaccurate.




On Thu, Jun 25, 2020 at 12:37 AM Jean-Luc Manguin <
jean-luc.manguin at unicaen.fr> wrote:

> Again, I refer to the definition of polyominoes (
> https://en.wikipedia.org/wiki/Polyomino) : "A polyomino is a plane
> geometric figure formed by joining one or more equal squares edge to edge."
>
> "One or more", this is clear... (unless someone can prove that zero is
> more than one)
>
> I agree that they should cross-refer (and that is another reason to put
> the same offset value).
>
> JLM
>
> ----- Mail original -----
> De: "Fred Lunnon" <fred.lunnon at gmail.com>
> ?: "seqfan" <seqfan at list.seqfan.eu>
> Envoy?: Mercredi 24 Juin 2020 17:51:37
> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset
> problem
>
> A000105 with offset = 0  is correct as it currently stands:
> there is 1 empty polyomino with  #cells = 0 .
>
>   My preference would be to attach an extra 1 at the front of both
> A000988 and A001168 , and edit to  offset = 0.
>
>   All three should cross-refer to the others; and similar remarks
> may apply to related sequences, such as A030228.   NJAS ?
>
> Fred Lunnon
>
>
> On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> > Hello,
> >
> > Everytime I edit a sequence, I receive justified remarks about my
> > non-respect of some rules, although of course this non-respect is not
> done
> > by purpose.
> >
> > In return, I would like to find in OEIS a care about coherence ; this
> > example needs (IMO) a little correction :
> > - A000105 has an offset = 0
> > - A000988 and A001168 have offset = 1
> >
> > These three sequences give the number of polyominoes (free, one-sided,
> and
> > fixed) and as it is written in the dedicated page in Mathworld, [
> > https://mathworld.wolfram.com/Polyomino.html |
> > https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino
> with
> > zero cell ; then I would be pleased to see A000105 with offset = 1, which
> > seems much more logical. Of course I could edit this sequence (I did not
> > try, I guess I can), but I prefer to mention this in the list.
> >
> > Thank you, dear editors, for your attention to this message.
> > Have a nice day and a happy St John !
> >
> > JL Manguin
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From Brendan.McKay at anu.edu.au  Thu Jun 25 11:15:55 2020
From: Brendan.McKay at anu.edu.au (Brendan McKay)
Date: Thu, 25 Jun 2020 19:15:55 +1000
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
Message-ID: <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>

Hi Neil,

Anyone who sees an error on Wikipedia, and knows how to fix it but 
doesn't, is
responsible for the error.

But in this case there is no error.? Wikipedia follows "reliable 
sources" and in this case
the reliable source is Golumb's book "Polyominoes".? From a brief look 
it seems that
Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and 
D.1), so
Wikipedia is *required* to either also disallow it or to change to a 
different source.

Incidentally, the first comment on A0000105, which is
"A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the 
moment.

Cheers, Brendan.

On 25/6/20 5:27 pm, Neil Sloane wrote:
> Everything in the Wikipedia is inaccurate.


From jean-luc.manguin at unicaen.fr  Thu Jun 25 14:15:46 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Thu, 25 Jun 2020 14:15:46 +0200 (CEST)
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
Message-ID: <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>

Hello everyone,

Thank you Brendan to "bring some water to my mill", and I am sorry to "shake the tree up to the top", but a request (actually 3) on OEIS with "number of fixed/one-sided/free polyominoes" give results that show this question is not definitely clear.
Moreover, the sequences such as A292357 could be strangely impacted if we admit that there is a "zeromino" ; in that case, we should complete the first diagonal with a "1" at the beginning (if I follow your hypothesis), but what should we put in the corresponding line and row ?? And what could mean "a polyomino with height = 0 and width = n" ??
And moreover again, why should we consider there is ONE zeromino ? Why not zero ?

The usual "rule" is (IMO) to exhibit a publication which justify such a choice ; till now, I have seen nothing.

Thank you for your attention.
Best regards,

JLM 

----- Mail original -----
De: "Brendan McKay" <Brendan.McKay at anu.edu.au>
?: "seqfan" <seqfan at list.seqfan.eu>
Envoy?: Jeudi 25 Juin 2020 11:15:55
Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset problem

Hi Neil,

Anyone who sees an error on Wikipedia, and knows how to fix it but 
doesn't, is
responsible for the error.

But in this case there is no error.? Wikipedia follows "reliable 
sources" and in this case
the reliable source is Golumb's book "Polyominoes".? From a brief look 
it seems that
Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and 
D.1), so
Wikipedia is *required* to either also disallow it or to change to a 
different source.

Incidentally, the first comment on A0000105, which is
"A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the 
moment.

Cheers, Brendan.

On 25/6/20 5:27 pm, Neil Sloane wrote:
> Everything in the Wikipedia is inaccurate.

--
Seqfan Mailing list - http://list.seqfan.eu/




From acwacw at gmail.com  Thu Jun 25 07:13:47 2020
From: acwacw at gmail.com (Allan Wechsler)
Date: Thu, 25 Jun 2020 01:13:47 -0400
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
Message-ID: <CADy-sGHW6P96fiAU=T9JnavBVkxpetCG_p=qN4q2vcQmhGu7Lw@mail.gmail.com>

If the question is, "Does OEIS acknowledge Wikipedia as the ultimate
arbiter of the definition of mathematical terms?", then I am fairly sure
that the answer is "no".

There are many, many sensible reasons to admit the empty set of cells as a
0-omino, and few serious reasons not to.

On Thu, Jun 25, 2020 at 12:37 AM Jean-Luc Manguin <
jean-luc.manguin at unicaen.fr> wrote:

> Again, I refer to the definition of polyominoes (
> https://en.wikipedia.org/wiki/Polyomino) : "A polyomino is a plane
> geometric figure formed by joining one or more equal squares edge to edge."
>
> "One or more", this is clear... (unless someone can prove that zero is
> more than one)
>
> I agree that they should cross-refer (and that is another reason to put
> the same offset value).
>
> JLM
>
> ----- Mail original -----
> De: "Fred Lunnon" <fred.lunnon at gmail.com>
> ?: "seqfan" <seqfan at list.seqfan.eu>
> Envoy?: Mercredi 24 Juin 2020 17:51:37
> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset
> problem
>
> A000105 with offset = 0  is correct as it currently stands:
> there is 1 empty polyomino with  #cells = 0 .
>
>   My preference would be to attach an extra 1 at the front of both
> A000988 and A001168 , and edit to  offset = 0.
>
>   All three should cross-refer to the others; and similar remarks
> may apply to related sequences, such as A030228.   NJAS ?
>
> Fred Lunnon
>
>
> On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> > Hello,
> >
> > Everytime I edit a sequence, I receive justified remarks about my
> > non-respect of some rules, although of course this non-respect is not
> done
> > by purpose.
> >
> > In return, I would like to find in OEIS a care about coherence ; this
> > example needs (IMO) a little correction :
> > - A000105 has an offset = 0
> > - A000988 and A001168 have offset = 1
> >
> > These three sequences give the number of polyominoes (free, one-sided,
> and
> > fixed) and as it is written in the dedicated page in Mathworld, [
> > https://mathworld.wolfram.com/Polyomino.html |
> > https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino
> with
> > zero cell ; then I would be pleased to see A000105 with offset = 1, which
> > seems much more logical. Of course I could edit this sequence (I did not
> > try, I guess I can), but I prefer to mention this in the list.
> >
> > Thank you, dear editors, for your attention to this message.
> > Have a nice day and a happy St John !
> >
> > JL Manguin
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From mail at oscarcunningham.com  Thu Jun 25 15:38:36 2020
From: mail at oscarcunningham.com (Oscar Cunningham)
Date: Thu, 25 Jun 2020 14:38:36 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
 <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
Message-ID: <6ab82fcb-1dfb-a8fe-a9a2-914ec7e891c4@oscarcunningham.com>

Hi all,

One of the defining features of polyominos is that they're connected. 
But I believe the correct convention is that the empty set is not 
connected. This is for the same reason that 1 is not a prime number. 
Good expositions of this idea are found in Harary and Read's 'Is the 
null-graph a pointless concept?' 
(https://link.springer.com/chapter/10.1007/BFb0066433) and the nLab page 
'too simple to be simple' 
(https://ncatlab.org/nlab/show/too+simple+to+be+simple).

So we should say that there is no polyomino of size 0.

Best,

Oscar Cunningham

On 25/06/2020 13:15, Jean-Luc Manguin wrote:
> Hello everyone,
>
> Thank you Brendan to "bring some water to my mill", and I am sorry to "shake the tree up to the top", but a request (actually 3) on OEIS with "number of fixed/one-sided/free polyominoes" give results that show this question is not definitely clear.
> Moreover, the sequences such as A292357 could be strangely impacted if we admit that there is a "zeromino" ; in that case, we should complete the first diagonal with a "1" at the beginning (if I follow your hypothesis), but what should we put in the corresponding line and row ?? And what could mean "a polyomino with height = 0 and width = n" ??
> And moreover again, why should we consider there is ONE zeromino ? Why not zero ?
>
> The usual "rule" is (IMO) to exhibit a publication which justify such a choice ; till now, I have seen nothing.
>
> Thank you for your attention.
> Best regards,
>
> JLM
>
> ----- Mail original -----
> De: "Brendan McKay" <Brendan.McKay at anu.edu.au>
> ?: "seqfan" <seqfan at list.seqfan.eu>
> Envoy?: Jeudi 25 Juin 2020 11:15:55
> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset problem
>
> Hi Neil,
>
> Anyone who sees an error on Wikipedia, and knows how to fix it but
> doesn't, is
> responsible for the error.
>
> But in this case there is no error.? Wikipedia follows "reliable
> sources" and in this case
> the reliable source is Golumb's book "Polyominoes".? From a brief look
> it seems that
> Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and
> D.1), so
> Wikipedia is *required* to either also disallow it or to change to a
> different source.
>
> Incidentally, the first comment on A0000105, which is
> "A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the
> moment.
>
> Cheers, Brendan.
>
> On 25/6/20 5:27 pm, Neil Sloane wrote:
>> Everything in the Wikipedia is inaccurate.
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/


From jack.grahl at gmail.com  Thu Jun 25 12:49:41 2020
From: jack.grahl at gmail.com (Jack Grahl)
Date: Thu, 25 Jun 2020 11:49:41 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
Message-ID: <CABC07KdKc7JNDm0xQgSNxivxN0W=QYhXsPRVsY-wPiA5EQSVDg@mail.gmail.com>

Notwithstanding what Wikipedia should do here, I think that the OEIS should
always include a zero term when this has basic mathematical validity (ie
for most combinatorial objects - providing that an instance with zero
elements can be described without contradiction). This makes the
encyclopedia much more useful, since relating a sequence to other
definitions, generating functions, and so on is a basic use of the
encyclopedia. The 'offset' info makes it easy to neglect these terms if you
want to.

On Thu, 25 Jun 2020, 10:16 Brendan McKay, <Brendan.McKay at anu.edu.au> wrote:

> Hi Neil,
>
> Anyone who sees an error on Wikipedia, and knows how to fix it but
> doesn't, is
> responsible for the error.
>
> But in this case there is no error.  Wikipedia follows "reliable
> sources" and in this case
> the reliable source is Golumb's book "Polyominoes".  From a brief look
> it seems that
> Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and
> D.1), so
> Wikipedia is *required* to either also disallow it or to change to a
> different source.
>
> Incidentally, the first comment on A0000105, which is
> "A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the
> moment.
>
> Cheers, Brendan.
>
> On 25/6/20 5:27 pm, Neil Sloane wrote:
> > Everything in the Wikipedia is inaccurate.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From christianperfect at gmail.com  Thu Jun 25 14:25:22 2020
From: christianperfect at gmail.com (Christian Lawson-Perfect)
Date: Thu, 25 Jun 2020 13:25:22 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
 <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
Message-ID: <CAEMHSOhhdCqsKjoKtQ6ah96YmQUvsSxX-26D2BqPcq9osebcrQ@mail.gmail.com>

This feels like the discussion about whether empty matrices exist. See "An
empty exercise" by Carl de Boor -
https://dl.acm.org/doi/10.1145/122272.122273

On Thu, 25 Jun 2020 at 13:15, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr>
wrote:

> Hello everyone,
>
> Thank you Brendan to "bring some water to my mill", and I am sorry to
> "shake the tree up to the top", but a request (actually 3) on OEIS with
> "number of fixed/one-sided/free polyominoes" give results that show this
> question is not definitely clear.
> Moreover, the sequences such as A292357 could be strangely impacted if we
> admit that there is a "zeromino" ; in that case, we should complete the
> first diagonal with a "1" at the beginning (if I follow your hypothesis),
> but what should we put in the corresponding line and row ?? And what could
> mean "a polyomino with height = 0 and width = n" ??
> And moreover again, why should we consider there is ONE zeromino ? Why not
> zero ?
>
> The usual "rule" is (IMO) to exhibit a publication which justify such a
> choice ; till now, I have seen nothing.
>
> Thank you for your attention.
> Best regards,
>
> JLM
>
> ----- Mail original -----
> De: "Brendan McKay" <Brendan.McKay at anu.edu.au>
> ?: "seqfan" <seqfan at list.seqfan.eu>
> Envoy?: Jeudi 25 Juin 2020 11:15:55
> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset
> problem
>
> Hi Neil,
>
> Anyone who sees an error on Wikipedia, and knows how to fix it but
> doesn't, is
> responsible for the error.
>
> But in this case there is no error.  Wikipedia follows "reliable
> sources" and in this case
> the reliable source is Golumb's book "Polyominoes".  From a brief look
> it seems that
> Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and
> D.1), so
> Wikipedia is *required* to either also disallow it or to change to a
> different source.
>
> Incidentally, the first comment on A0000105, which is
> "A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the
> moment.
>
> Cheers, Brendan.
>
> On 25/6/20 5:27 pm, Neil Sloane wrote:
> > Everything in the Wikipedia is inaccurate.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From jean-paul.allouche at imj-prg.fr  Thu Jun 25 19:14:54 2020
From: jean-paul.allouche at imj-prg.fr (jean-paul allouche)
Date: Thu, 25 Jun 2020 19:14:54 +0200
Subject: [seqfan] empty polyomino?
Message-ID: <0e1b8584-0bcf-7011-76eb-d15c81e83ca9@imj-prg.fr>

Hi

If a publication is a justification (I of course agree), open 
/scholar.google.com/,
type "empty polyomino" (with quotes), and contemplate...
An amusing point is that one of the paper says: "where we assume that w is
the contour of a non-empty polyomino" :-)

jean-paul


From fred.lunnon at gmail.com  Thu Jun 25 21:12:53 2020
From: fred.lunnon at gmail.com (Fred Lunnon)
Date: Thu, 25 Jun 2020 20:12:53 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAEMHSOhhdCqsKjoKtQ6ah96YmQUvsSxX-26D2BqPcq9osebcrQ@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
 <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
 <CAEMHSOhhdCqsKjoKtQ6ah96YmQUvsSxX-26D2BqPcq9osebcrQ@mail.gmail.com>
Message-ID: <CAN57YqufbtJoTZ4qW3ykDgcHGHA3x_PM4cq5B7sHiTU+YTwmZg@mail.gmail.com>

  Or indeed,
    https://en.wikipedia.org/wiki/Field_with_one_element

  Such questions may prove highly nontrivial, and their detailed investigation
reveal subtle disparities between apparently equivalent concepts, as well as
unexpected connections between apparently distinct ones.

  Didactically convenient simplified responses should therefore be avoided
--- but just don't mention  b*n*m*a* c*e*f*c*e*t*  !

WFL


On 6/25/20, Christian Lawson-Perfect <christianperfect at gmail.com> wrote:
> This feels like the discussion about whether empty matrices exist. See "An
> empty exercise" by Carl de Boor -
> https://dl.acm.org/doi/10.1145/122272.122273
>
> On Thu, 25 Jun 2020 at 13:15, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr>
> wrote:
>
>> Hello everyone,
>>
>> Thank you Brendan to "bring some water to my mill", and I am sorry to
>> "shake the tree up to the top", but a request (actually 3) on OEIS with
>> "number of fixed/one-sided/free polyominoes" give results that show this
>> question is not definitely clear.
>> Moreover, the sequences such as A292357 could be strangely impacted if we
>> admit that there is a "zeromino" ; in that case, we should complete the
>> first diagonal with a "1" at the beginning (if I follow your hypothesis),
>> but what should we put in the corresponding line and row ?? And what could
>> mean "a polyomino with height = 0 and width = n" ??
>> And moreover again, why should we consider there is ONE zeromino ? Why not
>> zero ?
>>
>> The usual "rule" is (IMO) to exhibit a publication which justify such a
>> choice ; till now, I have seen nothing.
>>
>> Thank you for your attention.
>> Best regards,
>>
>> JLM
>>
>> ----- Mail original -----
>> De: "Brendan McKay" <Brendan.McKay at anu.edu.au>
>> ?: "seqfan" <seqfan at list.seqfan.eu>
>> Envoy?: Jeudi 25 Juin 2020 11:15:55
>> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset
>> problem
>>
>> Hi Neil,
>>
>> Anyone who sees an error on Wikipedia, and knows how to fix it but
>> doesn't, is
>> responsible for the error.
>>
>> But in this case there is no error.  Wikipedia follows "reliable
>> sources" and in this case
>> the reliable source is Golumb's book "Polyominoes".  From a brief look
>> it seems that
>> Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and
>> D.1), so
>> Wikipedia is *required* to either also disallow it or to change to a
>> different source.
>>
>> Incidentally, the first comment on A0000105, which is
>> "A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the
>> moment.
>>
>> Cheers, Brendan.
>>
>> On 25/6/20 5:27 pm, Neil Sloane wrote:
>> > Everything in the Wikipedia is inaccurate.
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From tomaszordowski at gmail.com  Wed Jun 24 13:14:22 2020
From: tomaszordowski at gmail.com (Tomasz Ordowski)
Date: Wed, 24 Jun 2020 13:14:22 +0200
Subject: [seqfan] Related to Wolstenholme's theorem
In-Reply-To: <CAF0qcNOzrv_sRqP9Tsw+hhTEQgJ_obJk0EzTW9MS8_4z=2ymfw@mail.gmail.com>
References: <CAF0qcNOzrv_sRqP9Tsw+hhTEQgJ_obJk0EzTW9MS8_4z=2ymfw@mail.gmail.com>
Message-ID: <CAF0qcNP=ocrhjnMswk_0P9kk_9ELay_g8cRCM_yj57hJ+yf_hw@mail.gmail.com>

Dear readers!

Let F(n) = Sum_{k=1..n} 2^{k-1}/k = N(n)/D(n):
1/1, 2/1, 10/3, 16/3, 128/15, 208/15, 2416/105, ...
Theorem: If p > 3 is prime, then p F(p) == 1 (mod p^3);
equivalently p N(p) == D(p) (mod p^4), since p | D(p).
Conjecture: For n > 3, if n N(n) == D(n) (mod n^4),
then n is prime. If so, there are no such pseudoprimes.
Are there the weak pseudoprimes, except n = 49 ?
Composites n such that n N(n) == D(n) (mod n^3):
7^2, 16843^2, 2124679^2, ... To be confirmed.
Primes p such that p N(p) == D(p) (mod p^5)
are 7, 16843, ?  Cf. A088164:
https://oeis.org/A088164
Is p = 2124679 as well?

Best regards,

Thomas Ordowski
_______________________
https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem
     Note that N(n) = A108866(n)/2 : https://oeis.org/A108866
and D(2n+1) = D(2n+2) = A229726(n) : https://oeis.org/A229726


From jean-luc.manguin at unicaen.fr  Tue Jun  2 16:51:34 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Tue, 2 Jun 2020 16:51:34 +0200 (CEST)
Subject: [seqfan] A property of numbers in A171640
Message-ID: <1606652363.12731015.1591109494544.JavaMail.zimbra@unicaen.fr>

Hello, 

I would like to add a comment to the serie [ https://oeis.org/A171640 | A171640 ] 
But as I have 3 pending drafts I can't edit this sequence... 
Anyway I post it here. 
If a(n) is in A171240, then the number N = 6 x a(n) x (a(n) -1) is a square number. 
Moreover, if we consider the number P = 2 x (a(n) - 1) + sqrt(N) then : 

P x P + (P+1)x(P+1) + (P+2)x(P+2) = (P + a(n))x(P + a(n)) + (P + a(n) +1)x(P + a(n) +1) 

The first term a(1)=3 gives P=10 
and 10*10 + 11*11 + 12*12 = 13*13 + 14*14 = 365 

Thank you for your attention. 
Best regards. 

JL Manguin 



From mlb at well.com  Tue Jun  2 19:12:28 2020
From: mlb at well.com (Marc LeBrun)
Date: Tue, 2 Jun 2020 10:12:28 -0700
Subject: [seqfan] Re: A property of numbers in A171640
In-Reply-To: <1606652363.12731015.1591109494544.JavaMail.zimbra@unicaen.fr>
References: <1606652363.12731015.1591109494544.JavaMail.zimbra@unicaen.fr>
Message-ID: <026AEAD4-6BF2-4CDB-9039-B5BA06191A02@well.com>

>= Jean-Luc Manguin <jean-luc.manguin at unicaen.fr>
> If a(n) is in A171240...

Did you mean A171640 here?



From sven-h.simon at gmx.de  Wed Jun  3 21:29:31 2020
From: sven-h.simon at gmx.de (Sven Simon)
Date: Wed, 3 Jun 2020 21:29:31 +0200
Subject: [seqfan] Re: Websites on OEIS server
In-Reply-To: <CAAOnSgTxEOw_tG19WTtU85t2wTJ6-42BSFy=hr4ADCz71fiWgw@mail.gmail.com>
References: <000001d634fb$26ab3b40$7401b1c0$@gmx.de>
 <CAAOnSgSCDPvDHLSesk5WMScG78d+ebASvWfJwsPqtg8GdAD1mw@mail.gmail.com>
 <000001d63662$e8225f70$b8671e50$@gmx.de>
 <CAAOnSgTxEOw_tG19WTtU85t2wTJ6-42BSFy=hr4ADCz71fiWgw@mail.gmail.com>
Message-ID: <002d01d639dd$4ef2d0e0$ecd872a0$@gmx.de>

Hello Neil,
There was another email to the topic which did not make it into our emails.
I was at another location for a short period and did send the email another
way. This is it:
Hello Neil,
unforunately I do not have any experience in programming web pages. I think
it would be to much work to migrate any of the links included in the web
page and some of them are dead already either. If you only take the text it
is less work I estimate and one has more motivation to use the original
page. It is not our intention to remove the original page.
What I could do as a workaround is making screenshots of graphics and
interesting tables and we could place the jpg-files on the OEIS server and
link them to the sequence.
 
Sven

-----Urspr?ngliche Nachricht-----
Von: SeqFan <seqfan-bounces at list.seqfan.eu> Im Auftrag von Neil Sloane
Gesendet: Samstag, 30. Mai 2020 21:22
An: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc: Achim Flammenkamp <achim at uni-bielefeld.de>
Betreff: [seqfan] Re: Websites on OEIS server

Sven,  Thank you for that reply.  I will have to study this problem more
carefully, but I can't do it right now.  Very very busy.

I will try to do it soon.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sat, May 30, 2020 at 5:15 AM Sven Simon <sven-h.simon at gmx.de> wrote:

> Hello Neil,
> After a closer look on Achim Flammenkamp's webpage on multiperfect 
> numbers I think it is best to give you all that work of putting it to 
> the OEIS server. It does not make much sense if I make a Html file  of 
> it, and there will be this or that missing or it is not in the right 
> format while you can easily access it and make necessary changes. 
> Almost certainly not all the links included are suitable to have on 
> OEIS server. Might be we do not need any of the links at all. For 
> example there is a search function to search in the numbers list, that 
> would be a lot of work to migrate and it is not so urgent if you have 
> the complete data file anyway. Of course parts of the webpage describing
the search function would be not necessary then too.
>
> The page itself without further links is useful already in my opinion, 
> it has some interesting lists and graphics.
>
> The link to his multiperfect numbers page is
>
> http://wwwhomes.uni-bielefeld.de/achim/mpn.html
>
> OEIS sequence is A091443.
>
> The link to his page is already at the OEIS  sequence.
> His permission is to make a copy of his multiperfect numbers page 
> (link
> above) to place it on OEIS server, name it so it is clear it is a copy 
> of his page and add the date of the copy in the name. The link already 
> at the sequence shall remain, so when there are changes on his page at 
> Uni Bielefeld, we get these too. And one can use his additional 
> functionality on the original webpage.
>
> For those who speak German, that is his permission statement:
> Machen Sie ihre physische Kopie, stellen Sie diese oeffentlich, ABER 
> packen sie einfach eine Anmerkung/Note das dies eine Kpoie der 
> LINK-auf-original URL vom 2020-XX-YY ist.
> Dann kann jeder der mag auf den Link klicken und wird, falls sie oder 
> gar eine neuere Version dort ist diese sehen.
>
> Thanks
> Sven
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Seqfan Mailing list - http://list.seqfan.eu/



From alonso.delarte at gmail.com  Fri Jun  5 03:56:02 2020
From: alonso.delarte at gmail.com (Alonso Del Arte)
Date: Thu, 4 Jun 2020 21:56:02 -0400
Subject: [seqfan] Haverbeke's 1 plus 5 or times 3 problem
Message-ID: <CAGyGvfVCZ-9bRWaT4PkDuRUTzo7Db0h=G-6_jBd4-h-CT0DDbg@mail.gmail.com>

To illustrate recursive functions in *Eloquent JavaScript*, Marijn
Haverbeke posits the following problem: start from 1, then either add 5 or
multiply by 3. Repeat one or the other or alternate in whatever pattern you
want. Some numbers, like 13, can be reached in at least one way. Others,
like 15, are unreachable. Write a JavaScript function to find a solution
for a given number; it doesn't have to be the "best" solution, it simply
has to be *a* solution, if it exists for the particular number.

Maybe Haverbeke's the first to connect this mathematical problem to
JavaScript, but I doubt he's the one who first formulated it. Although it's
also possible that he remembered a similar problem and intentionally or
unintentionally changed it to suit his purpose. Though I doubt anyone
before me has explored the variant starting with ?1, which I'm not ready to
say anything more about at this time.

As usual for problems of this sort, the first reference to consult is the
OEIS. My searches got lots of results, but in this instance I suspect they
were often irrelevant matches, and a demonstration of the law of small
numbers.

Haverbeke bibliographic citation and link to Scastie snippet currently at
https://oeis.org/draft/A335365

My question to y'all: have you ever heard of this problem before, or a
similar problem?

Al

-- 
Alonso del Arte
Author at SmashWords.com
<https://www.smashwords.com/profile/view/AlonsoDelarte>
Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>


From njasloane at gmail.com  Fri Jun  5 04:40:35 2020
From: njasloane at gmail.com (Neil Sloane)
Date: Thu, 4 Jun 2020 22:40:35 -0400
Subject: [seqfan] Re: Haverbeke's 1 plus 5 or times 3 problem
In-Reply-To: <CAGyGvfVCZ-9bRWaT4PkDuRUTzo7Db0h=G-6_jBd4-h-CT0DDbg@mail.gmail.com>
References: <CAGyGvfVCZ-9bRWaT4PkDuRUTzo7Db0h=G-6_jBd4-h-CT0DDbg@mail.gmail.com>
Message-ID: <CAAOnSgTdvkqQiAuseTEB93xNZewdHXrih7g4WsNGNG5em2ij0Q@mail.gmail.com>

This is a type of sequence where we already have a number of examples
(although not that one). For example:

%N A121538 Increasing sequence: "if n appears then a*n+b doesn't", case
a(1)=1, a=2, b=1.
%C A121539 Equivalently, increasing sequence defined by: "if n appears
a*n+b does not", case a(1)=0, a=2, b=1.
%N A121540 Increasing sequence: "if n appears a*n+b does not", case a(1)=3,
a=2, b=1.
%N A121541 Increasing sequence: "if n appears a*n+b does not", case a(1)=4,
a=2, b=1.
%N A121542 Increasing sequence: "if n appears a*n+b does not", case a(1)=5,
a=2, b=1.
%C A003159 If n appears then 2n does not.
%N A005658 If n appears so do 2n, 3n+2, 6n+3.
%N A005659 If n appears so do 2n-2 and 3n-3.
%N A005660 If n appears so do 2n+2 and 3n+3.
%N A005662 If n appears then so do 2n+2 and 3n+3.
%C A022342 If n appears, n + (rank of n) does not (10 is the 7th term in
the sequence but 10 + 7 = 17 is not a term of the sequence). - _Benoit
Cloitre_, Jun 18 2002
%C A028260 If n appears, p*n does not (p primes). - _Philippe Del?ham_, Jun
10 2006
%C A036668 If n appears then 2n and 3n do not. - _Benoit Cloitre_, Jun 13
2002
%N A121537 If n appears then 2n, 3n and 4n do not.
%N A121543 "If n appears then n-th prime doesn't", with a(1)=1.

Besides your A335365, could you also submit "If n appears then so do n+5
and 3*n",  for completeness, even though it is very dense ?
I guess it begins
1, 3, 6, 8, 9, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, ...


On Thu, Jun 4, 2020 at 9:56 PM Alonso Del Arte <alonso.delarte at gmail.com>
wrote:

> To illustrate recursive functions in *Eloquent JavaScript*, Marijn
> Haverbeke posits the following problem: start from 1, then either add 5 or
> multiply by 3. Repeat one or the other or alternate in whatever pattern you
> want. Some numbers, like 13, can be reached in at least one way. Others,
> like 15, are unreachable. Write a JavaScript function to find a solution
> for a given number; it doesn't have to be the "best" solution, it simply
> has to be *a* solution, if it exists for the particular number.
>
> Maybe Haverbeke's the first to connect this mathematical problem to
> JavaScript, but I doubt he's the one who first formulated it. Although it's
> also possible that he remembered a similar problem and intentionally or
> unintentionally changed it to suit his purpose. Though I doubt anyone
> before me has explored the variant starting with ?1, which I'm not ready to
> say anything more about at this time.
>
> As usual for problems of this sort, the first reference to consult is the
> OEIS. My searches got lots of results, but in this instance I suspect they
> were often irrelevant matches, and a demonstration of the law of small
> numbers.
>
> Haverbeke bibliographic citation and link to Scastie snippet currently at
> https://oeis.org/draft/A335365
>
> My question to y'all: have you ever heard of this problem before, or a
> similar problem?
>
> Al
>
> --
> Alonso del Arte
> Author at SmashWords.com
> <https://www.smashwords.com/profile/view/AlonsoDelarte>
> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From jean-luc.manguin at unicaen.fr  Fri Jun  5 14:06:46 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Fri, 5 Jun 2020 14:06:46 +0200 (CEST)
Subject: [seqfan] Two questions about A287151 and A292357
Message-ID: <861407980.15411096.1591358806081.JavaMail.zimbra@unicaen.fr>

Hello, 

After considering A287151 and A292357, I have two questions : 

1) These series count "fixed" structures, I mean they do not take in account possible rotations or symmetries ; I did not manage to find their equivalent for "one-sided" structures nor for "free" structures (except A268371, which is much less developped). Does anyone know if this kind of sequence exist in OEIS ? 

2) These series count the total of structures in a m x n rectangle, but I would like to find detailed information about the number of cells (or elements) in the structures, as in A054252 for the squares, and A226048 for 2 x n rectangles, etc... ; my question is similar to the first one : does anyone know if this kind of sequence exist in OEIS ? 

In addition : I have data, but I try to search sequences starting from my data, and I got nothing ; anyway I prefer to ask first because they might exist in a different format (table read by antidiagonal, for instance). 

Thank you by advance for the answers. 
Best regards. 

JL Manguin 



From wouter.meeussen at telenet.be  Sun Jun  7 20:16:16 2020
From: wouter.meeussen at telenet.be (Wouter Meeussen)
Date: Sun, 7 Jun 2020 20:16:16 +0200
Subject: [seqfan] interesting idea, weird ramifications, as yet unsolved
Message-ID: <D9FBE905A7AB491997B0748B761F565B@Wou>

?Pairwise combinations of distinct elements?
see https://math.stackexchange.com/questions/3708151

my feeling is that Algebraic Combinatorics should clarify this, reducing it to a well known pattern.
I tried but failed dismally.

It boils down to a list of monomials X_(k+1) generated by

(X_k ** X_k  - X_k . X_k )/2 where ?**? represents the outer product.

The author (?Cesare?) only asks for a very limited property of these multinomials, but there is a lot of other ?enumerative combinatorics? going on here.

Does anyone recognise the patten?

Wouter.


From jmachacek.math at gmail.com  Mon Jun  8 04:23:28 2020
From: jmachacek.math at gmail.com (John Machacek)
Date: Sun, 7 Jun 2020 22:23:28 -0400
Subject: [seqfan] Re: questions about walks in the plane
In-Reply-To: <CA+Tsd+34_u-R_yxYFB78W0bLF6vfg+MEk4+gsLNaA4r9CULnPA@mail.gmail.com>
References: <CAAOnSgQCWJN1z9ctomfAJsrBbMAZNetrSXXCxs28RrnSEL01vA@mail.gmail.com>
 <MN2PR02MB6079AF46BEA2D848C28B11AAD98C0@MN2PR02MB6079.namprd02.prod.outlook.com>
 <CA+Tsd+3_XgU-LPs6k=+=yRAF4uaSWx1M+qnTET8+6+kdMLR1WA@mail.gmail.com>
 <CA+Tsd+34_u-R_yxYFB78W0bLF6vfg+MEk4+gsLNaA4r9CULnPA@mail.gmail.com>
Message-ID: <CA+Tsd+3xQbPteeZJgwTQd3EDCwYQFoLw7nB5ORu5p8WieF+1xQ@mail.gmail.com>

Hello,

I can explain the following

%C A085363 Apparently, the number of 2-D directed walks of semilength n
starting at (0,0) and ending on the X-axis using steps NE, SE, NW and SW
avoiding adjacent NW/SE and adjacent NE/SW. - _David Scambler_, Jun 20 2013

I come up with a new formula. I wasn't able to count the walks directly
with anything in A085363. Let s(n) be the number of such 2-D directed walks
of semilength n satisfying the conditions. I claim

s(n) = Sum_{k=1..n} [2^(2*n-2*k+2)*binomial(n-1,k-1)^2 +
2^(2*n-2*k+3)*binomial(n-1,k-1)*binomial(n-1,k-2)].

Then using Zeilberger's Algorithm we find that n*s(n) = (10*n-6)*s(n-1) -
(9*n-18)*s(n-2) which is the same recursion as for a(n) in A085363. We can
check the initial values match. Therefore s(n) = a(n) = A085363(n).

To see why the s(n) formula is correct let's look at an example.

Let's look at the term 2^(2n-2k+2)*binomial(n-1,k-1)*(binomial(n-1,k-1)
with n=4 and k = 2. Here choose two integer compositions of n each with k
parts. For example, (3,1) and (2,2). We then interleave to make a word
UUUDDUDD where U means up and D means down.

UUU (D)D (U) (D)D

Now with the exception of the letters in parentheses each U can be replaced
with NE or NW and similarly each D can be replaced with SE or SW. The
letters in parentheses are forced. There are 2^5 = 2^(8 - 4 + 1) choices.
The extra factor of 2 comes from switching roles of U and D (i.e.
reflecting path over axis).

The term 2^(2*n-2*k+3)*binomial(n-1,k-1)*binomial(n-1,k-2) is similar
except you have compositions like (2,2) and (4) so a word like UUDDDDUU.

We also get two hypergeometric evaluations.

s(n) = 4^n 3F2(-n, 1+n, 1-n; 1, n; 1/4)
s(n) = (8n - 4)* 3F2(2-2n, 1-n, 1-n; 1-2n, 2; 4)

Best,
John Machacek

On Sun, May 31, 2020 at 12:10 PM John Machacek <jmachacek.math at gmail.com>
wrote:

> ...sorry typo in my previous email on the initial conditions. Of course
> a(2) = 1 since the empty walk is the unique walk of length zero.
>
> On Sun, May 31, 2020 at 12:08 PM John Machacek <jmachacek.math at gmail.com>
> wrote:
>
>> Hello,
>>
>> For A001630 the walks are "ternary" words in {-1, +1, +3}, but the
>> "length" is sum of the absolute values (as opposed to length as a word in
>> {-1, +1, +3}).
>>
>> So, a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) since
>>
>> a(n-1) accounts for walks ending with (+1)
>> a(n-2) accounts for walks ending with (+1, -1)
>> a(n-3) accounts for walks ending with (+3)
>> a(n-4) accounts for walks ending with (+3, -1).
>>
>> Then we check the initial conditions also work
>>
>> a(2) = 0: (empty)
>> a(3) = 2: (+1), (-1)
>> a(4) = 3: (+1,+1), (+1,-1), (-1,+1)
>> a(5) = 6: (+1,+1,+1), (+1,+1,-1), (+1,-1,+1), (-1,+1,+1), (-1,+1,-1), (+3)
>>
>> I don't currently have anything to say about the 2-D walks. But I'll try
>> to think about them...
>>
>> Best,
>> John Machacek
>>
>>
>> On Sun, May 31, 2020 at 3:25 AM Nacin, David <NACIND at wpunj.edu> wrote:
>>
>>> Though the last three questions are clear, I'm confused on the first.
>>> If we are talking one-D walks using +1,-1, +3 with no consecutive -1's then
>>> there must be some other restriction as well, otherwise the sequence would
>>> just contain A028859<https://oeis.org/A028859>
>>> 1,3,8,22,60,164,448,1224,... .  (It also wouldn't matter what the numbers
>>> themselves were, only that one of the numbers can't be repeated
>>> consecutively.)  What am I missing?  Does the walk have to end at a certain
>>> value?
>>>
>>> -David
>>>
>>> ________________________________
>>> From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of Neil Sloane <
>>> njasloane at gmail.com>
>>> Sent: Thursday, May 28, 2020 12:26 PM
>>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>> Subject: [seqfan] questions about walks in the plane
>>>
>>> An old friend (Kees Immink) asked me about the conjecture of David
>>> Scambler
>>> in A085363.  In fact there are 4 assertions/conjectures in the OEIS of
>>> this
>>> type: (the first is only a one-D walk)
>>>
>>> %C A001630 Apparently for n>=2 the number of 1-D walks of length n-2
>>> using
>>> steps +1, +3 and -1, avoiding consecutive -1 steps. - _David Scambler_,
>>> Jul
>>> 15 2013
>>>
>>> %C A084768 Number of directed 2-D walks of length 2n starting at (0,0)
>>> and
>>> ending on the X-axis using steps NE, SE, NE, SW and avoiding NE followed
>>> by
>>> SE. - _David Scambler_, Jun 24 2013
>>>
>>> %C A085363 Apparently, the number of 2-D directed walks of semilength n
>>> starting at (0,0) and ending on the X-axis using steps NE, SE, NW and SW
>>> avoiding adjacent NW/SE and adjacent NE/SW. - _David Scambler_, Jun 20
>>> 2013
>>>
>>> %C A101500 Directed 2-D walks with n steps starting at (0,0) and ending
>>> on
>>> the X-axis using steps N,S,E,W and avoiding N followed by S. - _David
>>> Scambler_, Jun 24 2013
>>>
>>> I know we have several experts here - could someone help and provide
>>> proofs?
>>>
>>> The third question is the following:
>>> Let a(n) = the number of 2-D directed walks of semilength n starting at
>>> (0,0) and ending on the X-axis using steps NE, SE, NW and SW avoiding
>>> adjacent NW/SE and adjacent NE/SW
>>> Show that this satisfies the recurrence
>>> a(0)=1; for n>0: a(n) = 4*9^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).
>>>
>>> (The second and fourth questions are stated as if they are theorems, but
>>> no
>>> proof is given.)
>>>
>>> Neil
>>>
>>> --
>>> Seqfan Mailing list -
>>> https://nam11.safelinks.protection.outlook.com/?url=http%3A%2F%2Flist.seqfan.eu%2F&amp;data=01%7C01%7C%7Ce198304f85fa4fff06d108d804520b8a%7C74540637643546cc87a46d38efb78538%7C0&amp;sdata=DzmnJsOnapeI6w15s1%2FMrnFBFPcQvW%2FCkFOw6mLibDY%3D&amp;reserved=0
>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>


From pemd70 at yahoo.com  Sun Jun  7 21:52:17 2020
From: pemd70 at yahoo.com (Ali Sada)
Date: Sun, 7 Jun 2020 19:52:17 +0000 (UTC)
Subject: [seqfan] Connection between A000245 and A026016 (or A071725)?
References: <791425054.629861.1591559537899.ref@mail.yahoo.com>
Message-ID: <791425054.629861.1591559537899@mail.yahoo.com>

Hi Everyone,

Arrange the numbers between 1 and n in a raw of size 2n such that: 
a) 1 is always in the first spot;
b) k's position in the raw is <= 2k; and ?
c) k always comes before k+1


For example, below are the possibilities we get when n = 3. (I put ?*? instead of a space for more clarity.) 

1, 2, 3, *,* ,* 
1, 2, *, 3, *, *
1, 2, *, *, 3, * 
1, 2, *, *, *, 3
1, *, 2, 3, *, * 
1, *, 2, *, 3, *
1, *, 2, *, *, 3
1, *, *, 2, 3, * 
1, *, *, 2, *, 3

The possibilities I got were 1, 3, 9, 28, 90, 297, 1001, 3432, 11934 (A000245.)

Then, I filled the remaining spaces with the numbers between n+1 and 2n. 

Using n = 3 as an example we get: 

1, 2, 3, 4, 5, 6
1, 2, 4, 3, 5, 6
1, 2, 4, 5, 3, 6
1, 2, 4, 5, 6, 3
1, 4, 2, 3, 5, 6
1, 4, 2, 5, 3, 6
1, 4, 2, 5, 6, 3
1, 4, 5, 2, 3, 6
1, 4, 5, 2, 6, 3

For the numbers I tested (only 8,) the sum of the third column minus the sum of the second column was A026016(n-1).

In the example of n = 3 above, the sum of the third column is 31, and the sum of the second column is 28. 
31-28 = 3, which is A026016(2).

3 is also the result of the subtraction of the fourth column minus the third column, and the sixth column minus the fifth column. 

I would really appreciate it if you could tell me if this pattern will continue and why.

Best,

Ali





From baruchel at gmx.com  Mon Jun  8 11:07:34 2020
From: baruchel at gmx.com (Thomas Baruchel)
Date: Mon, 8 Jun 2020 11:07:34 +0200 (CEST)
Subject: [seqfan] Looking for a function expanding to (1-q^k)^(-1) terms
Message-ID: <alpine.DEB.2.11.2006081104480.374@raspberrypi>

Hi,

is someone aware of some function expanding to the following series:


     x/(1-q)  +  x^2 / (1-q^2)   +  x^3 / (1-q^3)   + ...

I have a close match with the QPolyGamma function which evaluates to the
sum above at x=1, but I am not sure I can fully build the given series from it.

Thank you by advance,

best regards,

--
Thomas Baruchel


From simon.plouffe at gmail.com  Mon Jun  8 11:34:54 2020
From: simon.plouffe at gmail.com (Simon Plouffe)
Date: Mon, 8 Jun 2020 11:34:54 +0200
Subject: [seqfan] Re: Looking for a function expanding to (1-q^k)^(-1) terms
In-Reply-To: <alpine.DEB.2.11.2006081104480.374@raspberrypi>
References: <alpine.DEB.2.11.2006081104480.374@raspberrypi>
Message-ID: <CAK+PtObiNVGWZSMdofAgek161SKU485p=oCBon6FANjSbrNvOg@mail.gmail.com>

hello Thomas,
but this is tau(n) * x^n, isn't ?
this is well known.
just expand the sum into a series. ...

Simon



Le lun. 8 juin 2020 ? 11:25, Thomas Baruchel <baruchel at gmx.com> a ?crit :

> Hi,
>
> is someone aware of some function expanding to the following series:
>
>
>      x/(1-q)  +  x^2 / (1-q^2)   +  x^3 / (1-q^3)   + ...
>
> I have a close match with the QPolyGamma function which evaluates to the
> sum above at x=1, but I am not sure I can fully build the given series
> from it.
>
> Thank you by advance,
>
> best regards,
>
> --
> Thomas Baruchel
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From techsubs at pearceneptune.co.uk  Mon Jun  8 18:01:07 2020
From: techsubs at pearceneptune.co.uk (Peter Munn)
Date: Mon, 8 Jun 2020 17:01:07 +0100
Subject: [seqfan] A107008 = primes of the form 24k+1?
Message-ID: <78b93da138ac4a07e4af5195cfb89d51.squirrel@webmail.nsnoc.com>

Hi seqfans.

In A107008 (Primes of the form x^2+24*y^2), NJAS comments "Presumably this
is the same as primes congruent to 1 mod 24."

Can we come up with something to decide this?

It would help to establish when 24k+1 can be written as x^2+24*y^2. I
reckon this happens when the k in 24k+1 can be written as the sum of a
square, j, and a generalized pentagonal number, i, because setting y =
sqrt(j), x = sqrt(24i+1) can be shown to satisfy the equation. (See Zak
Seidov's 2008 comment in http://oeis.org/A001318, "Generalized pentagonal
numbers".) I believe the converse is true, also.

OEIS does not yet have the sequence "Numbers that can't be written as the
sum of a square and a generalized pentagonal number" [1], but the
sequence, S, starts 20, 29, 33, 34, 45, 46, 53, ... , and I reckon it has
positive asymptotic density.

So the question becomes: if k is a term of S, why should 24k+1 be
composite, at least up to the limit of Vladimir Orlovsky's check in
A107008? From the first 3 terms we get: 20*24 + 1 = 481 = 13*37; 29*24 + 1
= 697 = 17*41; 33*24 + 1 = 793 = 13*61.

Going through more terms, I saw a pattern emerge, prompting me to ask: is
this particular subset of "24k+1" numbers the same as "nonsquare numbers
of the form (24i + m) * (24j + m), 0 <= i < j, m in {13, 17, 19, 23}"?
This would be interesting anyway, and could be a clue.

However, I'm not sure I'm close to an answer, and there might be a much
easier route: does anyone have better ideas? Or know the answer already?

Best regards,

Peter

[1] I also tried looking for the number of ways positive integers _can_ be
so written, using "Search: seq:2,2,1,1,2,2,1,1,2,1,2,1,1,1,1,4" but it
draws a blank.
























From techsubs at pearceneptune.co.uk  Mon Jun  8 20:30:14 2020
From: techsubs at pearceneptune.co.uk (Peter Munn)
Date: Mon, 8 Jun 2020 19:30:14 +0100
Subject: [seqfan] Re: A107008 = primes of the form 24k+1?
Message-ID: <05651d9a07c70d6a8df607d445185932.squirrel@webmail.nsnoc.com>

Following a private response on this subject, I should emphasize that the
tricky part seems to be to show that _all_ primes congruent to 1 mod 24
are in A107008. And after a little more spreadsheet work, it is starting
to look particularly interesting...

Does anyone want to check to what extent the following hypothesis is true?:

When k can be written as the sum of a square and a generalized pentagonal
number in exactly one way, the resulting numbers of the form 24k+1 might
be exactly the prime numbers of the form 24k+1 plus the squares of prime
numbers congruent to 13, 17, 19 or 23 mod 24.

Peter

On Mon, June 8, 2020 5:01 pm, Peter Munn wrote:
> Hi seqfans.
> In A107008 (Primes of the form x^2+24*y^2), NJAS comments "Presumably
this
> is the same as primes congruent to 1 mod 24."
> Can we come up with something to decide this?
> It would help to establish when 24k+1 can be written as x^2+24*y^2. I
reckon this happens when the k in 24k+1 can be written as the sum of a
square, j, and a generalized pentagonal number, i, because setting y =
sqrt(j), x = sqrt(24i+1) can be shown to satisfy the equation. (See Zak
Seidov's 2008 comment in http://oeis.org/A001318, "Generalized
pentagonal
> numbers".) I believe the converse is true, also.
> OEIS does not yet have the sequence "Numbers that can't be written as
the
> sum of a square and a generalized pentagonal number" [1], but the
sequence, S, starts 20, 29, 33, 34, 45, 46, 53, ... , and I reckon it
has
> positive asymptotic density.
> So the question becomes: if k is a term of S, why should 24k+1 be
composite, at least up to the limit of Vladimir Orlovsky's check in
A107008? From the first 3 terms we get: 20*24 + 1 = 481 = 13*37; 29*24 +
1
> = 697 = 17*41; 33*24 + 1 = 793 = 13*61.
> Going through more terms, I saw a pattern emerge, prompting me to ask:
is
> this particular subset of "24k+1" numbers the same as "nonsquare numbers
of the form (24i + m) * (24j + m), 0 <= i < j, m in {13, 17, 19, 23}"?
This would be interesting anyway, and could be a clue.
> However, I'm not sure I'm close to an answer, and there might be a much
easier route: does anyone have better ideas? Or know the answer already?
Best regards,
> Peter
> [1] I also tried looking for the number of ways positive integers _can_
be
> so written, using "Search: seq:2,2,1,1,2,2,1,1,2,1,2,1,1,1,1,4" but it
draws a blank.
> --
> Seqfan Mailing list - http://list.seqfan.eu/






From wouter.meeussen at telenet.be  Tue Jun  9 12:11:42 2020
From: wouter.meeussen at telenet.be (Wouter Meeussen)
Date: Tue, 9 Jun 2020 12:11:42 +0200
Subject: [seqfan] Re: interesting idea, weird ramifications, as yet unsolved
In-Reply-To: <1457329491.2128123.1591680009085@mail1.libero.it>
References: <D9FBE905A7AB491997B0748B761F565B@Wou>
 <1457329491.2128123.1591680009085@mail1.libero.it>
Message-ID: <A85A70B168864715B67DF39D26B97C60@Wou>

We can look at the question asked by 'Cesare' in a wider scope:

define xx(0) as the vector {x_1, ... , x_n} and define xx(1) as the outer 
product Sum(i=1..n,j=1..n;  x_i x_j ) , subtract the diagonal elements 
Sum(i=1..n ; x_i x_i) and divide by two. This extracts only the terms below 
the diagonal of the outer product.
Repeat this procedure up to xx(k).
The result is a symmetric polynomial in n variables of degree 2^k.

The number of terms (monomials) as function of n and k is :

n=3 ; 3,  3,   3,      3, ...
n=4 ; 4,  6, 15,  102,      5010,          12 367 734, 76 069 184 710 488, 
...
n=5 ; 5, 10, 45, 975, 470 025, 110 131 246 500, ...


for n=4, there are 12 monomials containing 3 variables, all others have 4 
variables.
for n=5, I find 30 monomials in 3 variables, rest in 4 or 5 variables
(all for k>1 of course)

nice detail:
the monomials in 3 variables are simple when written in terms of the 
monomial symmetric functions:
for all k and n=2,...,7 they are m({1,1}) , m({2,1,1}), m({3,3,2}), 
m({6,5,5}), m({11,11,10}), m({22,21,21}) etc.

Wouter.
(sanity checks are always appreciated)
__________________________________
in Mathematica:
Remove[X];n=4;
X[0] = Array[Subscript[x, #] &, n];
X[k_Integer /; k > 0] :=  X[k] =
List @@ Expand[(Apply[Plus, Flatten[Outer[Times, X[k - 1], X[k - 1]]]] - 
Dot[X[k - 1], X[k - 1]])/2 ];

__________________________________

-----Original Message----- 
From: Luca Petrone
Sent: Tuesday, June 09, 2020 7:20 AM
To: Sequence Fanatics Discussion list ; Wouter Meeussen
Subject: Re: [seqfan] interesting idea, weird ramifications, as yet unsolved

Of course there are "easy" formulas to calculate how many elements of X_N 
have, for example, k repetition of one element, but things get non-sensely 
complicated of you are asking k repetions of one item, l repetitions of 
another item and so on: is really this the question? and what is it the 
meaning of it?
Best Regards,
Luca
> Il 7 giugno 2020 alle 20.16 Wouter Meeussen <wouter.meeussen at telenet.be> 
> ha scritto:
>
>
> ?Pairwise combinations of distinct elements?
> see https://math.stackexchange.com/questions/3708151
>
> my feeling is that Algebraic Combinatorics should clarify this, reducing 
> it to a well known pattern.
> I tried but failed dismally.
>
> It boils down to a list of monomials X_(k+1) generated by
>
> (X_k ** X_k  - X_k . X_k )/2 where ?**? represents the outer product.
>
> The author (?Cesare?) only asks for a very limited property of these 
> multinomials, but there is a lot of other ?enumerative combinatorics? 
> going on here.
>
> Does anyone recognise the patten?
>
> Wouter.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/ 



From pemd70 at yahoo.com  Wed Jun 10 23:07:49 2020
From: pemd70 at yahoo.com (Ali Sada)
Date: Wed, 10 Jun 2020 21:07:49 +0000 (UTC)
Subject: [seqfan] All-Fives Domino Sequence
References: <1975428622.2422057.1591823269175.ref@mail.yahoo.com>
Message-ID: <1975428622.2422057.1591823269175@mail.yahoo.com>

Hi Everyone,

In Iraq, we play a variation of the All-Five game called Aznif. (Some scholars say the name comes of an Armenian word that means ?noble lady?). 

These are the rules:

1)?? ?You can start with any tile.
2)?? ?The first double on the ground becomes the (only) spinner.
3)?? ?The goal is to score as many multiples of 5 by adding the numbers on the ends in each step. (The points on the spinner don?t count after it is covered from 2 sides).


In this variation, we can score 470 points using all 28 tiles of the order 6 domino set. These pictures show how, step by step: 

https://justpaste.it/2ipxj

I don't have a proof, but I think we can?t score more than 470.

The sequence here is the total points we can score using a domino set of order n.

For, n = 0, we have only one tile 0?0, and we can?t score anything. So, a(0) = 0.
For, n = 1, we have three tiles, and we can?t score anything. So, a(1) = 0.
For n = 2, we have six tiles. We can only score 5 once. So, a(2) = 5.
For n = 3, we have ten tiles. We can score, 5+5+10+10+10+10. So, a(3) = 50.

And so on.
The question is: can we find a way to count the scores for domino sets with higher orders (other than trial and error)?
Also, is this sequence suitable for the OEIS?

Best,

Ali




From luca.petrone at libero.it  Tue Jun  9 07:20:09 2020
From: luca.petrone at libero.it (Luca Petrone)
Date: Tue, 9 Jun 2020 07:20:09 +0200 (CEST)
Subject: [seqfan] Re: interesting idea, weird ramifications, as yet unsolved
In-Reply-To: <D9FBE905A7AB491997B0748B761F565B@Wou>
References: <D9FBE905A7AB491997B0748B761F565B@Wou>
Message-ID: <1457329491.2128123.1591680009085@mail1.libero.it>

Of course there are "easy" formulas to calculate how many elements of X_N have, for example, k repetition of one element, but things get non-sensely complicated of you are asking k repetions of one item, l repetitions of another item and so on: is really this the question? and what is it the meaning of it?
Best Regards,
Luca
> Il 7 giugno 2020 alle 20.16 Wouter Meeussen <wouter.meeussen at telenet.be> ha scritto:
> 
> 
> ?Pairwise combinations of distinct elements?
> see https://math.stackexchange.com/questions/3708151
> 
> my feeling is that Algebraic Combinatorics should clarify this, reducing it to a well known pattern.
> I tried but failed dismally.
> 
> It boils down to a list of monomials X_(k+1) generated by
> 
> (X_k ** X_k  - X_k . X_k )/2 where ?**? represents the outer product.
> 
> The author (?Cesare?) only asks for a very limited property of these multinomials, but there is a lot of other ?enumerative combinatorics? going on here.
> 
> Does anyone recognise the patten?
> 
> Wouter.
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/


From bradklee at gmail.com  Wed Jun 17 04:43:26 2020
From: bradklee at gmail.com (Brad Klee)
Date: Tue, 16 Jun 2020 21:43:26 -0500
Subject: [seqfan] Ramanujan-Inspired Sequences for 1/pi.
Message-ID: <CAC_LFVasF-og0ChFyh4XNf_DRufXBkYDsTx+uAx1PyaeVpm+Uw@mail.gmail.com>

s=2: 1, 6, 56, 620, 7512, 96208, 1279168, 17471448, 243509720, 3447792656,
49434765888 . . .   nAn.

s=3: 1, 9, 138, 2550, 51840, 1116612, 24999408, 575368596, 13518747000,
322765065480, 7805239515216 . . .   nAn.

s=4: 1, 18, 632, 27300, 1306200, 66413424, 3515236032, 191434588488,
10648603594200, 602109586993200, 34491343330027584 . . .   nAn.

s=6: 1, 90, 20280, 5798100, 1854085464, 632693421360, 225235329359040,
82598530506097320, 30962429500615006680, 11803615010304909757680,
4560219108801622243897920 . . .   nAn.

They are all double binomial sums, but can be reduced to single sums using
Zeilberger's algorithm. Dividing a(n) by c^n for c=32,54,128, or 864,
summing
over n, we obtain either 4/pi, 9*sqrt(3)/(4*pi), 8*sqrt(2)/(3*pi) or
18/(5*pi).

These are a fairly straightforward consequence of the assertion in S.14 of:
http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf

More on this soon...

--Brad


From bpcrtz at free.fr  Mon Jun  1 14:14:56 2020
From: bpcrtz at free.fr (Paul Curtz)
Date: Mon, 01 Jun 2020 12:14:56 -0000
Subject: [seqfan] Pseudotriangle
In-Reply-To: <1647911946.153700598.1591006184939.JavaMail.root@spooler7-g27.priv.proxad.net>
Message-ID: <1834687573.153960332.1591013696293.JavaMail.root@spooler7-g27.priv.proxad.net>

Hello,
                                   sum
   0                                0
   0                                0
   0  1                             1
   0  2  2                          4
   0  3  4                          7
   0  4  6  3                      13
   0  5  8  6  4                   23
   0  6 10  9  8                   33
   0  7 12 12 12  5                48
   0  8 14 15 16 10  6             69
   0  9 16 18 20 15 12             90
   ...
  b(n) = 0, 0, A038391(n).

b(n) first two differences:
  0,  0,  1,  4,  7, 13, 23, 33, 48, 69, 90, 118, 154, ...
  0,  1,  3,  3,  6, 10, 10, 15, 21, 21, 28,  36,  36, ... (from A000217(n))
  1,  2,  0,  3,  4,  0,  5,  6,  0,  7,  8,   0,   9, ... = A117909(n).

Is this kind of partial triangle known?

Best regards

Paul


From pemd70 at yahoo.com  Fri Jun 19 04:10:01 2020
From: pemd70 at yahoo.com (Ali Sada)
Date: Fri, 19 Jun 2020 02:10:01 +0000 (UTC)
Subject: [seqfan] Four grid-based sequences
References: <401142798.530443.1592532601323.ref@mail.yahoo.com>
Message-ID: <401142798.530443.1592532601323@mail.yahoo.com>

Hi Everyone,

Please see the 4 sequences below. I need help with their definitions and with finding more terms. Sorry in advance if the terminology I am using is not proper.? 

1.?? ?In the corner of a 2D grid, we put 1. Now, we put the other numbers such that n is connected to n-1. Symmetry is not allowed.
 We start with a(1) = 1. 

Now, 2 has 2 potential squares: (1,2) and (2,2). So, a(2) = 2.
Based on the places of 2, 3 will have 8 potential squares.
4 will have 38 potential squares, and 5 will have 196, and so on.
1 ,2 , 8, 38, 196, 

These shapes might explain the idea in a better way: 
https://justpaste.it/6qxc0


2.?? ?The same idea of the first sequence except that n could be connected to any number already on the grid. Symmetry is not allowed.
The sequence here will be 1, 2, 8, 53, 463, 

These are the shapes generated by this algorithm ?
https://justpaste.it/6qju4

3.?? ?The same idea of the first sequence but the grid here has no edges. We put 1 in any square on the grid and we continue. n should be connected to n-1, and symmetry is not allowed. 

The sequence here is
1, 2, 8, 47, 295,? 

These are the shapes generated by this algorithm ?
https://justpaste.it/4l3p4


4.?? ?In this version, the grid here also has no edges, n could be connected to any number already on the grid, and symmetry is not allowed. 

The sequence here is 
1, 2, 13, 154, 2419,
These are the shapes generated by this algorithm 
https://justpaste.it/2b92w
(I know it seems like self-serving, but I really like the shapes of this version!)

I found the terms of the sequences above ?by hand?, so the possibility of a mistake is there, as usual.

I also need help with two more sequences. The same ideas of sequences 2 and 4, but with one symbol (e.g. ?*?) instead of numbers. 

Best,

Ali





















From jean-luc.manguin at unicaen.fr  Fri Jun 19 09:37:11 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Fri, 19 Jun 2020 09:37:11 +0200 (CEST)
Subject: [seqfan] Re: Four grid-based sequences
In-Reply-To: <401142798.530443.1592532601323@mail.yahoo.com>
References: <401142798.530443.1592532601323.ref@mail.yahoo.com>
 <401142798.530443.1592532601323@mail.yahoo.com>
Message-ID: <368101878.7333177.1592552231063.JavaMail.zimbra@unicaen.fr>

Hello Ali,

Your problem is (IMO) a variation of the problem of polyominoes ; the differences are :
- the nodes are labelled with numbers.
- corner connections are allowed

So the model to use in this case is graph-based, with labelled nodes.

The labels make things a bit more complicated, but as you also consider symmetry or not, it can be solved by a backtracking algorithm (transfer matrix algo is much faster but symmetry is difficult to take in account, maybe impossible...)
If I had to solve it, I would consider the problem without labels and make an implementation of the algorithm. After that you could add the constraints with the labels.

Best regards,

JLM

----- Mail original -----
De: "seqfan" <seqfan at list.seqfan.eu>
?: "seqfan" <seqfan at list.seqfan.eu>
Cc: "Ali Sada" <pemd70 at yahoo.com>
Envoy?: Vendredi 19 Juin 2020 04:10:01
Objet: [seqfan] Four grid-based sequences

Hi Everyone,

Please see the 4 sequences below. I need help with their definitions and with finding more terms. Sorry in advance if the terminology I am using is not proper.? 

1.?? ?In the corner of a 2D grid, we put 1. Now, we put the other numbers such that n is connected to n-1. Symmetry is not allowed.
 We start with a(1) = 1. 

Now, 2 has 2 potential squares: (1,2) and (2,2). So, a(2) = 2.
Based on the places of 2, 3 will have 8 potential squares.
4 will have 38 potential squares, and 5 will have 196, and so on.
1 ,2 , 8, 38, 196, 

These shapes might explain the idea in a better way: 
https://justpaste.it/6qxc0


2.?? ?The same idea of the first sequence except that n could be connected to any number already on the grid. Symmetry is not allowed.
The sequence here will be 1, 2, 8, 53, 463, 

These are the shapes generated by this algorithm ?
https://justpaste.it/6qju4

3.?? ?The same idea of the first sequence but the grid here has no edges. We put 1 in any square on the grid and we continue. n should be connected to n-1, and symmetry is not allowed. 

The sequence here is
1, 2, 8, 47, 295,? 

These are the shapes generated by this algorithm ?
https://justpaste.it/4l3p4


4.?? ?In this version, the grid here also has no edges, n could be connected to any number already on the grid, and symmetry is not allowed. 

The sequence here is 
1, 2, 13, 154, 2419,
These are the shapes generated by this algorithm 
https://justpaste.it/2b92w
(I know it seems like self-serving, but I really like the shapes of this version!)

I found the terms of the sequences above ?by hand?, so the possibility of a mistake is there, as usual.

I also need help with two more sequences. The same ideas of sequences 2 and 4, but with one symbol (e.g. ?*?) instead of numbers. 

Best,

Ali




















--
Seqfan Mailing list - http://list.seqfan.eu/




From sairvin at gmail.com  Mon Jun 22 03:51:20 2020
From: sairvin at gmail.com (Sean A. Irvine)
Date: Mon, 22 Jun 2020 13:51:20 +1200
Subject: [seqfan] A032435 and A032436 Josephus problem data mismatch
Message-ID: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>

Hi all,

The triangles in A032434 through A032436 concern the Josephus problem.
That's the problem where n people stand in a circle and then you go round
and round killing every k-th person until nearly everyone is dead.

The data in A032435 and A032436 do not match what I expect.

There are links from these sequences into Mathworld.  The Mathworld entry
contains two triangles which do contain the data I expect and which also
point back to these OEIS entries.  Neither of the triangles in the
Mathworld entry actually seem to be present in the OEIS.

However, we don't simply want to replace the existing data in A032435 and
A032436, since the current data values are likely related to this problem
somehow.  I spent a couple of hours looking at the current data values and
could not determine an explanation, perhaps someone else will have more
success.

Sean.


From njasloane at gmail.com  Mon Jun 22 04:52:31 2020
From: njasloane at gmail.com (Neil Sloane)
Date: Sun, 21 Jun 2020 22:52:31 -0400
Subject: [seqfan] Re: A032435 and A032436 Josephus problem data mismatch
In-Reply-To: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>
References: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>
Message-ID: <CAAOnSgTJcoe2xiffY-s4J-=h_Q9RDNLszLOz30Wc67ANeVAdew@mail.gmail.com>

The refs for A032435 are Ball-Coxeter, and one of Kraitchik's books.  I
checked Ball-Coxeter, but it was not helpful (though I only skimmed it).
I have three of Kraitchik's books, but apparently not that one.  So that
would be the next place to check.  Maybe it is online somewhere?

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sun, Jun 21, 2020 at 9:51 PM Sean A. Irvine <sairvin at gmail.com> wrote:

> Hi all,
>
> The triangles in A032434 through A032436 concern the Josephus problem.
> That's the problem where n people stand in a circle and then you go round
> and round killing every k-th person until nearly everyone is dead.
>
> The data in A032435 and A032436 do not match what I expect.
>
> There are links from these sequences into Mathworld.  The Mathworld entry
> contains two triangles which do contain the data I expect and which also
> point back to these OEIS entries.  Neither of the triangles in the
> Mathworld entry actually seem to be present in the OEIS.
>
> However, we don't simply want to replace the existing data in A032435 and
> A032436, since the current data values are likely related to this problem
> somehow.  I spent a couple of hours looking at the current data values and
> could not determine an explanation, perhaps someone else will have more
> success.
>
> Sean.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From amiram.eldar at gmail.com  Mon Jun 22 07:18:47 2020
From: amiram.eldar at gmail.com (Ami Eldar)
Date: Mon, 22 Jun 2020 08:18:47 +0300
Subject: [seqfan] Re: A032435 and A032436 Josephus problem data mismatch
In-Reply-To: <CAAOnSgTJcoe2xiffY-s4J-=h_Q9RDNLszLOz30Wc67ANeVAdew@mail.gmail.com>
References: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>
 <CAAOnSgTJcoe2xiffY-s4J-=h_Q9RDNLszLOz30Wc67ANeVAdew@mail.gmail.com>
Message-ID: <CAMJ5+vfhTtRSCoW1BM1zJiDLJ6+-CYyt=MNDBhsu-wYN5YuWQQ@mail.gmail.com>

The book "Mathematical Recreations" by  Kraitchik can be borrowed (freely
browsed for a limited time) at the Internet Archive:
https://archive.org/details/mathematicalrecr0002krai
https://archive.org/details/mathematicalrecr0000krai
https://archive.org/details/mathematicalrecr00krai



On Mon, Jun 22, 2020 at 5:52 AM Neil Sloane <njasloane at gmail.com> wrote:

> The refs for A032435 are Ball-Coxeter, and one of Kraitchik's books.  I
> checked Ball-Coxeter, but it was not helpful (though I only skimmed it).
> I have three of Kraitchik's books, but apparently not that one.  So that
> would be the next place to check.  Maybe it is online somewhere?
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Sun, Jun 21, 2020 at 9:51 PM Sean A. Irvine <sairvin at gmail.com> wrote:
>
> > Hi all,
> >
> > The triangles in A032434 through A032436 concern the Josephus problem.
> > That's the problem where n people stand in a circle and then you go round
> > and round killing every k-th person until nearly everyone is dead.
> >
> > The data in A032435 and A032436 do not match what I expect.
> >
> > There are links from these sequences into Mathworld.  The Mathworld entry
> > contains two triangles which do contain the data I expect and which also
> > point back to these OEIS entries.  Neither of the triangles in the
> > Mathworld entry actually seem to be present in the OEIS.
> >
> > However, we don't simply want to replace the existing data in A032435 and
> > A032436, since the current data values are likely related to this problem
> > somehow.  I spent a couple of hours looking at the current data values
> and
> > could not determine an explanation, perhaps someone else will have more
> > success.
> >
> > Sean.
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From olivier.gerard at gmail.com  Mon Jun 22 08:24:10 2020
From: olivier.gerard at gmail.com (Olivier Gerard)
Date: Mon, 22 Jun 2020 09:24:10 +0300
Subject: [seqfan] Re: A032435 and A032436 Josephus problem data mismatch
In-Reply-To: <CAMJ5+vfhTtRSCoW1BM1zJiDLJ6+-CYyt=MNDBhsu-wYN5YuWQQ@mail.gmail.com>
References: <CAL0FETnUXwLSbpU_okwsQshYKPtQ=1QWQTA2Ob++CT+hX1mTGg@mail.gmail.com>
 <CAAOnSgTJcoe2xiffY-s4J-=h_Q9RDNLszLOz30Wc67ANeVAdew@mail.gmail.com>
 <CAMJ5+vfhTtRSCoW1BM1zJiDLJ6+-CYyt=MNDBhsu-wYN5YuWQQ@mail.gmail.com>
Message-ID: <CAAcpPSbmtHq62EBYipF1NiOpE6V=L_3FmjAOxhyC4kUP+xCJww@mail.gmail.com>

Thanks to Jean-Paul Allouche who found it by other means, I have
now forwarded the relevant page to all interested parties.

If there are other people wanting to read it, I will forward it to them.

Olivier G?rard
Seqfan Mailing List Administrator



On Mon, Jun 22, 2020 at 9:00 AM Ami Eldar <amiram.eldar at gmail.com> wrote:

> The book "Mathematical Recreations" by  Kraitchik can be borrowed (freely
> browsed for a limited time) at the Internet Archive:
> https://archive.org/details/mathematicalrecr0002krai
> https://archive.org/details/mathematicalrecr0000krai
> https://archive.org/details/mathematicalrecr00krai
>
>
>
> On Mon, Jun 22, 2020 at 5:52 AM Neil Sloane <njasloane at gmail.com> wrote:
>
> > The refs for A032435 are Ball-Coxeter, and one of Kraitchik's books.  I
> > checked Ball-Coxeter, but it was not helpful (though I only skimmed it).
> > I have three of Kraitchik's books, but apparently not that one.  So that
> > would be the next place to check.  Maybe it is online somewhere?
> >
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, President, OEIS Foundation.
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > Email: njasloane at gmail.com
> >
> >
> >
> > On Sun, Jun 21, 2020 at 9:51 PM Sean A. Irvine <sairvin at gmail.com>
> wrote:
> >
> > > Hi all,
> > >
> > > The triangles in A032434 through A032436 concern the Josephus problem.
> > > That's the problem where n people stand in a circle and then you go
> round
> > > and round killing every k-th person until nearly everyone is dead.
> > >
> > > The data in A032435 and A032436 do not match what I expect.
> > >
> > > There are links from these sequences into Mathworld.  The Mathworld
> entry
> > > contains two triangles which do contain the data I expect and which
> also
> > > point back to these OEIS entries.  Neither of the triangles in the
> > > Mathworld entry actually seem to be present in the OEIS.
> > >
> > > However, we don't simply want to replace the existing data in A032435
> and
> > > A032436, since the current data values are likely related to this
> problem
> > > somehow.  I spent a couple of hours looking at the current data values
> > and
> > > could not determine an explanation, perhaps someone else will have more
> > > success.
> > >
> > > Sean.
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From DavidRabahy at comcast.net  Tue Jun 23 20:40:13 2020
From: DavidRabahy at comcast.net (David Rabahy)
Date: Tue, 23 Jun 2020 14:40:13 -0400
Subject: [seqfan] like A004080 but for 1/(2i) and more
Message-ID: <001901d6498d$bbad4700$3307d500$@comcast.net>

Least k such that H2(k) >= n, where H2(k) is the even harmonic number sum_{i=1..k} 1/(2i);
n
1 - 5
2 - 31
3 - 227
4 - 1674
...

Is this worthy of an entry in the OEIS or too trivial?

https://docs.google.com/spreadsheets/d/1GbRABHLdAtFHU0dFDkHqanJYje15L3gbc6RrWzWCnXo

Clearly this idea could be generalized, i.e. where Hq(k) is the qth harmonic number sum_{i=1..k} 1/(qi).

Also, it could be diagonalized;
n
1 - 1
2 - 31
3 - 4550
...



From seqfan at hasler.fr  Wed Jun 24 05:47:41 2020
From: seqfan at hasler.fr (M. F. Hasler)
Date: Tue, 23 Jun 2020 23:47:41 -0400
Subject: [seqfan] Re: like A004080 but for 1/(2i) and more
In-Reply-To: <001901d6498d$bbad4700$3307d500$@comcast.net>
References: <001901d6498d$bbad4700$3307d500$@comcast.net>
Message-ID: <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>

On Tue, Jun 23, 2020 at 10:33 PM David Rabahy <DavidRabahy at comcast.net>
wrote:

> Least k such that H2(k) >= n, where H2(k) is the even harmonic number
> sum_{i=1..k} 1/(2i);
> n
> 1 - 5
> 2 - 31
> 3 - 227
> 4 - 1674
>

Unless I'm wrong it should be 4, not 5, for n=1 above,
and more generally, A4080(2n), for obvious reasons :

Clearly this idea could be generalized, i.e. where Hq(k) is the qth
> harmonic number sum_{i=1..k} 1/(qi).
>

which is the same as H(k)/q, whence your sequence should be A4080(qn).

- Maximilian


From jean-luc.manguin at unicaen.fr  Wed Jun 24 15:20:03 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Wed, 24 Jun 2020 15:20:03 +0200 (CEST)
Subject: [seqfan] Polyominoes (A000105, A000988, A001168) - offset problem
Message-ID: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>

Hello, 

Everytime I edit a sequence, I receive justified remarks about my non-respect of some rules, although of course this non-respect is not done by purpose. 

In return, I would like to find in OEIS a care about coherence ; this example needs (IMO) a little correction : 
- A000105 has an offset = 0 
- A000988 and A001168 have offset = 1 

These three sequences give the number of polyominoes (free, one-sided, and fixed) and as it is written in the dedicated page in Mathworld, [ https://mathworld.wolfram.com/Polyomino.html | https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino with zero cell ; then I would be pleased to see A000105 with offset = 1, which seems much more logical. Of course I could edit this sequence (I did not try, I guess I can), but I prefer to mention this in the list. 

Thank you, dear editors, for your attention to this message. 
Have a nice day and a happy St John ! 

JL Manguin 



From fred.lunnon at gmail.com  Wed Jun 24 17:51:37 2020
From: fred.lunnon at gmail.com (Fred Lunnon)
Date: Wed, 24 Jun 2020 16:51:37 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
Message-ID: <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>

  A000105 with offset = 0  is correct as it currently stands:
there is 1 empty polyomino with  #cells = 0 .

  My preference would be to attach an extra 1 at the front of both
A000988 and A001168 , and edit to  offset = 0.

  All three should cross-refer to the others; and similar remarks
may apply to related sequences, such as A030228.   NJAS ?

Fred Lunnon


On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> Hello,
>
> Everytime I edit a sequence, I receive justified remarks about my
> non-respect of some rules, although of course this non-respect is not done
> by purpose.
>
> In return, I would like to find in OEIS a care about coherence ; this
> example needs (IMO) a little correction :
> - A000105 has an offset = 0
> - A000988 and A001168 have offset = 1
>
> These three sequences give the number of polyominoes (free, one-sided, and
> fixed) and as it is written in the dedicated page in Mathworld, [
> https://mathworld.wolfram.com/Polyomino.html |
> https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino with
> zero cell ; then I would be pleased to see A000105 with offset = 1, which
> seems much more logical. Of course I could edit this sequence (I did not
> try, I guess I can), but I prefer to mention this in the list.
>
> Thank you, dear editors, for your attention to this message.
> Have a nice day and a happy St John !
>
> JL Manguin
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From sairvin at gmail.com  Wed Jun 24 21:00:50 2020
From: sairvin at gmail.com (Sean A. Irvine)
Date: Thu, 25 Jun 2020 07:00:50 +1200
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
Message-ID: <CAL0FETkCXE48Aqsv-2_hcXds3NbA6UdB3Peh+Xvy8Tm=o14cAA@mail.gmail.com>

I'm with Fred on this.  The current values are all correct and consistent,
but indeed a(0)=1 could be inserted as Fred suggests.


On Thu, 25 Jun 2020 at 03:51, Fred Lunnon <fred.lunnon at gmail.com> wrote:

>   A000105 with offset = 0  is correct as it currently stands:
> there is 1 empty polyomino with  #cells = 0 .
>
>   My preference would be to attach an extra 1 at the front of both
> A000988 and A001168 , and edit to  offset = 0.
>
>   All three should cross-refer to the others; and similar remarks
> may apply to related sequences, such as A030228.   NJAS ?
>
> Fred Lunnon
>
>
> On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> > Hello,
> >
> > Everytime I edit a sequence, I receive justified remarks about my
> > non-respect of some rules, although of course this non-respect is not
> done
> > by purpose.
> >
> > In return, I would like to find in OEIS a care about coherence ; this
> > example needs (IMO) a little correction :
> > - A000105 has an offset = 0
> > - A000988 and A001168 have offset = 1
> >
> > These three sequences give the number of polyominoes (free, one-sided,
> and
> > fixed) and as it is written in the dedicated page in Mathworld, [
> > https://mathworld.wolfram.com/Polyomino.html |
> > https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino
> with
> > zero cell ; then I would be pleased to see A000105 with offset = 1, which
> > seems much more logical. Of course I could edit this sequence (I did not
> > try, I guess I can), but I prefer to mention this in the list.
> >
> > Thank you, dear editors, for your attention to this message.
> > Have a nice day and a happy St John !
> >
> > JL Manguin
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From rgwv at rgwv.com  Wed Jun 24 21:22:59 2020
From: rgwv at rgwv.com (rgwv at rgwv.com)
Date: Wed, 24 Jun 2020 15:22:59 -0400
Subject: [seqfan] Re: like A004080 but for 1/(2i) and more
In-Reply-To: <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>
References: <001901d6498d$bbad4700$3307d500$@comcast.net>
 <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>
Message-ID: <242501d64a5c$e0b763d0$a2262b70$@rgwv.com>

This is just the bisection of A002387.

-----Original Message-----
From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of M. F. Hasler
Sent: Tuesday, June 23, 2020 11:48 PM
To: DavidRabahy at comcast.net
Cc: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: like A004080 but for 1/(2i) and more

On Tue, Jun 23, 2020 at 10:33 PM David Rabahy <DavidRabahy at comcast.net>
wrote:

> Least k such that H2(k) >= n, where H2(k) is the even harmonic number 
> sum_{i=1..k} 1/(2i); n
> 1 - 5
> 2 - 31
> 3 - 227
> 4 - 1674
>

Unless I'm wrong it should be 4, not 5, for n=1 above,
and more generally, A4080(2n), for obvious reasons :

Clearly this idea could be generalized, i.e. where Hq(k) is the qth
> harmonic number sum_{i=1..k} 1/(qi).
>

which is the same as H(k)/q, whence your sequence should be A4080(qn).

- Maximilian

--
Seqfan Mailing list - http://list.seqfan.eu/



From njasloane at gmail.com  Wed Jun 24 21:47:55 2020
From: njasloane at gmail.com (Neil Sloane)
Date: Wed, 24 Jun 2020 15:47:55 -0400
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAL0FETkCXE48Aqsv-2_hcXds3NbA6UdB3Peh+Xvy8Tm=o14cAA@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <CAL0FETkCXE48Aqsv-2_hcXds3NbA6UdB3Peh+Xvy8Tm=o14cAA@mail.gmail.com>
Message-ID: <CAAOnSgQbwCbG7Ztd1sr38tp7eK=Pgu8NgMdKmXcBfwHZ99WxTg@mail.gmail.com>

I will follow Fred's advice, as always.


Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Wed, Jun 24, 2020 at 3:01 PM Sean A. Irvine <sairvin at gmail.com> wrote:

> I'm with Fred on this.  The current values are all correct and consistent,
> but indeed a(0)=1 could be inserted as Fred suggests.
>
>
> On Thu, 25 Jun 2020 at 03:51, Fred Lunnon <fred.lunnon at gmail.com> wrote:
>
> >   A000105 with offset = 0  is correct as it currently stands:
> > there is 1 empty polyomino with  #cells = 0 .
> >
> >   My preference would be to attach an extra 1 at the front of both
> > A000988 and A001168 , and edit to  offset = 0.
> >
> >   All three should cross-refer to the others; and similar remarks
> > may apply to related sequences, such as A030228.   NJAS ?
> >
> > Fred Lunnon
> >
> >
> > On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> > > Hello,
> > >
> > > Everytime I edit a sequence, I receive justified remarks about my
> > > non-respect of some rules, although of course this non-respect is not
> > done
> > > by purpose.
> > >
> > > In return, I would like to find in OEIS a care about coherence ; this
> > > example needs (IMO) a little correction :
> > > - A000105 has an offset = 0
> > > - A000988 and A001168 have offset = 1
> > >
> > > These three sequences give the number of polyominoes (free, one-sided,
> > and
> > > fixed) and as it is written in the dedicated page in Mathworld, [
> > > https://mathworld.wolfram.com/Polyomino.html |
> > > https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino
> > with
> > > zero cell ; then I would be pleased to see A000105 with offset = 1,
> which
> > > seems much more logical. Of course I could edit this sequence (I did
> not
> > > try, I guess I can), but I prefer to mention this in the list.
> > >
> > > Thank you, dear editors, for your attention to this message.
> > > Have a nice day and a happy St John !
> > >
> > > JL Manguin
> > >
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From jean-luc.manguin at unicaen.fr  Wed Jun 24 17:59:13 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Wed, 24 Jun 2020 17:59:13 +0200 (CEST)
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
Message-ID: <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>

Again, I refer to the definition of polyominoes (https://en.wikipedia.org/wiki/Polyomino) : "A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge."

"One or more", this is clear... (unless someone can prove that zero is more than one)

I agree that they should cross-refer (and that is another reason to put the same offset value).

JLM

----- Mail original -----
De: "Fred Lunnon" <fred.lunnon at gmail.com>
?: "seqfan" <seqfan at list.seqfan.eu>
Envoy?: Mercredi 24 Juin 2020 17:51:37
Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset problem

A000105 with offset = 0  is correct as it currently stands:
there is 1 empty polyomino with  #cells = 0 .

  My preference would be to attach an extra 1 at the front of both
A000988 and A001168 , and edit to  offset = 0.

  All three should cross-refer to the others; and similar remarks
may apply to related sequences, such as A030228.   NJAS ?

Fred Lunnon


On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> Hello,
>
> Everytime I edit a sequence, I receive justified remarks about my
> non-respect of some rules, although of course this non-respect is not done
> by purpose.
>
> In return, I would like to find in OEIS a care about coherence ; this
> example needs (IMO) a little correction :
> - A000105 has an offset = 0
> - A000988 and A001168 have offset = 1
>
> These three sequences give the number of polyominoes (free, one-sided, and
> fixed) and as it is written in the dedicated page in Mathworld, [
> https://mathworld.wolfram.com/Polyomino.html |
> https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino with
> zero cell ; then I would be pleased to see A000105 with offset = 1, which
> seems much more logical. Of course I could edit this sequence (I did not
> try, I guess I can), but I prefer to mention this in the list.
>
> Thank you, dear editors, for your attention to this message.
> Have a nice day and a happy St John !
>
> JL Manguin
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Seqfan Mailing list - http://list.seqfan.eu/




From davidrabahy at comcast.net  Wed Jun 24 19:25:36 2020
From: davidrabahy at comcast.net (davidrabahy at comcast.net)
Date: Wed, 24 Jun 2020 13:25:36 -0400
Subject: [seqfan] Re: like A004080 but for 1/(2i) and more
In-Reply-To: <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>
References: <001901d6498d$bbad4700$3307d500$@comcast.net>
 <CABxCbJ0ZuP2yY9d1iZ9p1q5FGG5X5aXoKaKe0V4SN_ErrZcYaA@mail.gmail.com>
Message-ID: <005801d64a4c$7d6eed80$784cc880$@comcast.net>

Of course you are correct; I?m not sure how I got myself confused.  A014537 is the sequence I was looking for.

 

Is the diagonalization 1,3,4550,? interesting enough for an OEIS entry?

 

From: M. F. Hasler <seqfan at hasler.fr> 
Sent: Tuesday, June 23, 2020 11:48 PM
To: DavidRabahy at comcast.net
Cc: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: Re: [seqfan] like A004080 but for 1/(2i) and more

 

On Tue, Jun 23, 2020 at 10:33 PM David Rabahy <DavidRabahy at comcast.net <mailto:DavidRabahy at comcast.net> > wrote:

Least k such that H2(k) >= n, where H2(k) is the even harmonic number sum_{i=1..k} 1/(2i);
n
1 - 5
2 - 31
3 - 227
4 - 1674

 

Unless I'm wrong it should be 4, not 5, for n=1 above,

and more generally, A4080(2n), for obvious reasons :

 

Clearly this idea could be generalized, i.e. where Hq(k) is the qth harmonic number sum_{i=1..k} 1/(qi).

 

which is the same as H(k)/q, whence your sequence should be A4080(qn).

 

- Maximilian

 



From njasloane at gmail.com  Thu Jun 25 09:27:00 2020
From: njasloane at gmail.com (Neil Sloane)
Date: Thu, 25 Jun 2020 03:27:00 -0400
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
Message-ID: <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>

Everything in the Wikipedia is inaccurate.




On Thu, Jun 25, 2020 at 12:37 AM Jean-Luc Manguin <
jean-luc.manguin at unicaen.fr> wrote:

> Again, I refer to the definition of polyominoes (
> https://en.wikipedia.org/wiki/Polyomino) : "A polyomino is a plane
> geometric figure formed by joining one or more equal squares edge to edge."
>
> "One or more", this is clear... (unless someone can prove that zero is
> more than one)
>
> I agree that they should cross-refer (and that is another reason to put
> the same offset value).
>
> JLM
>
> ----- Mail original -----
> De: "Fred Lunnon" <fred.lunnon at gmail.com>
> ?: "seqfan" <seqfan at list.seqfan.eu>
> Envoy?: Mercredi 24 Juin 2020 17:51:37
> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset
> problem
>
> A000105 with offset = 0  is correct as it currently stands:
> there is 1 empty polyomino with  #cells = 0 .
>
>   My preference would be to attach an extra 1 at the front of both
> A000988 and A001168 , and edit to  offset = 0.
>
>   All three should cross-refer to the others; and similar remarks
> may apply to related sequences, such as A030228.   NJAS ?
>
> Fred Lunnon
>
>
> On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> > Hello,
> >
> > Everytime I edit a sequence, I receive justified remarks about my
> > non-respect of some rules, although of course this non-respect is not
> done
> > by purpose.
> >
> > In return, I would like to find in OEIS a care about coherence ; this
> > example needs (IMO) a little correction :
> > - A000105 has an offset = 0
> > - A000988 and A001168 have offset = 1
> >
> > These three sequences give the number of polyominoes (free, one-sided,
> and
> > fixed) and as it is written in the dedicated page in Mathworld, [
> > https://mathworld.wolfram.com/Polyomino.html |
> > https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino
> with
> > zero cell ; then I would be pleased to see A000105 with offset = 1, which
> > seems much more logical. Of course I could edit this sequence (I did not
> > try, I guess I can), but I prefer to mention this in the list.
> >
> > Thank you, dear editors, for your attention to this message.
> > Have a nice day and a happy St John !
> >
> > JL Manguin
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From Brendan.McKay at anu.edu.au  Thu Jun 25 11:15:55 2020
From: Brendan.McKay at anu.edu.au (Brendan McKay)
Date: Thu, 25 Jun 2020 19:15:55 +1000
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
Message-ID: <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>

Hi Neil,

Anyone who sees an error on Wikipedia, and knows how to fix it but 
doesn't, is
responsible for the error.

But in this case there is no error.? Wikipedia follows "reliable 
sources" and in this case
the reliable source is Golumb's book "Polyominoes".? From a brief look 
it seems that
Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and 
D.1), so
Wikipedia is *required* to either also disallow it or to change to a 
different source.

Incidentally, the first comment on A0000105, which is
"A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the 
moment.

Cheers, Brendan.

On 25/6/20 5:27 pm, Neil Sloane wrote:
> Everything in the Wikipedia is inaccurate.


From jean-luc.manguin at unicaen.fr  Thu Jun 25 14:15:46 2020
From: jean-luc.manguin at unicaen.fr (Jean-Luc Manguin)
Date: Thu, 25 Jun 2020 14:15:46 +0200 (CEST)
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
Message-ID: <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>

Hello everyone,

Thank you Brendan to "bring some water to my mill", and I am sorry to "shake the tree up to the top", but a request (actually 3) on OEIS with "number of fixed/one-sided/free polyominoes" give results that show this question is not definitely clear.
Moreover, the sequences such as A292357 could be strangely impacted if we admit that there is a "zeromino" ; in that case, we should complete the first diagonal with a "1" at the beginning (if I follow your hypothesis), but what should we put in the corresponding line and row ?? And what could mean "a polyomino with height = 0 and width = n" ??
And moreover again, why should we consider there is ONE zeromino ? Why not zero ?

The usual "rule" is (IMO) to exhibit a publication which justify such a choice ; till now, I have seen nothing.

Thank you for your attention.
Best regards,

JLM 

----- Mail original -----
De: "Brendan McKay" <Brendan.McKay at anu.edu.au>
?: "seqfan" <seqfan at list.seqfan.eu>
Envoy?: Jeudi 25 Juin 2020 11:15:55
Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset problem

Hi Neil,

Anyone who sees an error on Wikipedia, and knows how to fix it but 
doesn't, is
responsible for the error.

But in this case there is no error.? Wikipedia follows "reliable 
sources" and in this case
the reliable source is Golumb's book "Polyominoes".? From a brief look 
it seems that
Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and 
D.1), so
Wikipedia is *required* to either also disallow it or to change to a 
different source.

Incidentally, the first comment on A0000105, which is
"A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the 
moment.

Cheers, Brendan.

On 25/6/20 5:27 pm, Neil Sloane wrote:
> Everything in the Wikipedia is inaccurate.

--
Seqfan Mailing list - http://list.seqfan.eu/




From acwacw at gmail.com  Thu Jun 25 07:13:47 2020
From: acwacw at gmail.com (Allan Wechsler)
Date: Thu, 25 Jun 2020 01:13:47 -0400
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
Message-ID: <CADy-sGHW6P96fiAU=T9JnavBVkxpetCG_p=qN4q2vcQmhGu7Lw@mail.gmail.com>

If the question is, "Does OEIS acknowledge Wikipedia as the ultimate
arbiter of the definition of mathematical terms?", then I am fairly sure
that the answer is "no".

There are many, many sensible reasons to admit the empty set of cells as a
0-omino, and few serious reasons not to.

On Thu, Jun 25, 2020 at 12:37 AM Jean-Luc Manguin <
jean-luc.manguin at unicaen.fr> wrote:

> Again, I refer to the definition of polyominoes (
> https://en.wikipedia.org/wiki/Polyomino) : "A polyomino is a plane
> geometric figure formed by joining one or more equal squares edge to edge."
>
> "One or more", this is clear... (unless someone can prove that zero is
> more than one)
>
> I agree that they should cross-refer (and that is another reason to put
> the same offset value).
>
> JLM
>
> ----- Mail original -----
> De: "Fred Lunnon" <fred.lunnon at gmail.com>
> ?: "seqfan" <seqfan at list.seqfan.eu>
> Envoy?: Mercredi 24 Juin 2020 17:51:37
> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset
> problem
>
> A000105 with offset = 0  is correct as it currently stands:
> there is 1 empty polyomino with  #cells = 0 .
>
>   My preference would be to attach an extra 1 at the front of both
> A000988 and A001168 , and edit to  offset = 0.
>
>   All three should cross-refer to the others; and similar remarks
> may apply to related sequences, such as A030228.   NJAS ?
>
> Fred Lunnon
>
>
> On 6/24/20, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr> wrote:
> > Hello,
> >
> > Everytime I edit a sequence, I receive justified remarks about my
> > non-respect of some rules, although of course this non-respect is not
> done
> > by purpose.
> >
> > In return, I would like to find in OEIS a care about coherence ; this
> > example needs (IMO) a little correction :
> > - A000105 has an offset = 0
> > - A000988 and A001168 have offset = 1
> >
> > These three sequences give the number of polyominoes (free, one-sided,
> and
> > fixed) and as it is written in the dedicated page in Mathworld, [
> > https://mathworld.wolfram.com/Polyomino.html |
> > https://mathworld.wolfram.com/Polyomino.html ] , there is no polyomino
> with
> > zero cell ; then I would be pleased to see A000105 with offset = 1, which
> > seems much more logical. Of course I could edit this sequence (I did not
> > try, I guess I can), but I prefer to mention this in the list.
> >
> > Thank you, dear editors, for your attention to this message.
> > Have a nice day and a happy St John !
> >
> > JL Manguin
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From mail at oscarcunningham.com  Thu Jun 25 15:38:36 2020
From: mail at oscarcunningham.com (Oscar Cunningham)
Date: Thu, 25 Jun 2020 14:38:36 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
 <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
Message-ID: <6ab82fcb-1dfb-a8fe-a9a2-914ec7e891c4@oscarcunningham.com>

Hi all,

One of the defining features of polyominos is that they're connected. 
But I believe the correct convention is that the empty set is not 
connected. This is for the same reason that 1 is not a prime number. 
Good expositions of this idea are found in Harary and Read's 'Is the 
null-graph a pointless concept?' 
(https://link.springer.com/chapter/10.1007/BFb0066433) and the nLab page 
'too simple to be simple' 
(https://ncatlab.org/nlab/show/too+simple+to+be+simple).

So we should say that there is no polyomino of size 0.

Best,

Oscar Cunningham

On 25/06/2020 13:15, Jean-Luc Manguin wrote:
> Hello everyone,
>
> Thank you Brendan to "bring some water to my mill", and I am sorry to "shake the tree up to the top", but a request (actually 3) on OEIS with "number of fixed/one-sided/free polyominoes" give results that show this question is not definitely clear.
> Moreover, the sequences such as A292357 could be strangely impacted if we admit that there is a "zeromino" ; in that case, we should complete the first diagonal with a "1" at the beginning (if I follow your hypothesis), but what should we put in the corresponding line and row ?? And what could mean "a polyomino with height = 0 and width = n" ??
> And moreover again, why should we consider there is ONE zeromino ? Why not zero ?
>
> The usual "rule" is (IMO) to exhibit a publication which justify such a choice ; till now, I have seen nothing.
>
> Thank you for your attention.
> Best regards,
>
> JLM
>
> ----- Mail original -----
> De: "Brendan McKay" <Brendan.McKay at anu.edu.au>
> ?: "seqfan" <seqfan at list.seqfan.eu>
> Envoy?: Jeudi 25 Juin 2020 11:15:55
> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset problem
>
> Hi Neil,
>
> Anyone who sees an error on Wikipedia, and knows how to fix it but
> doesn't, is
> responsible for the error.
>
> But in this case there is no error.? Wikipedia follows "reliable
> sources" and in this case
> the reliable source is Golumb's book "Polyominoes".? From a brief look
> it seems that
> Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and
> D.1), so
> Wikipedia is *required* to either also disallow it or to change to a
> different source.
>
> Incidentally, the first comment on A0000105, which is
> "A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the
> moment.
>
> Cheers, Brendan.
>
> On 25/6/20 5:27 pm, Neil Sloane wrote:
>> Everything in the Wikipedia is inaccurate.
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/


From jack.grahl at gmail.com  Thu Jun 25 12:49:41 2020
From: jack.grahl at gmail.com (Jack Grahl)
Date: Thu, 25 Jun 2020 11:49:41 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
Message-ID: <CABC07KdKc7JNDm0xQgSNxivxN0W=QYhXsPRVsY-wPiA5EQSVDg@mail.gmail.com>

Notwithstanding what Wikipedia should do here, I think that the OEIS should
always include a zero term when this has basic mathematical validity (ie
for most combinatorial objects - providing that an instance with zero
elements can be described without contradiction). This makes the
encyclopedia much more useful, since relating a sequence to other
definitions, generating functions, and so on is a basic use of the
encyclopedia. The 'offset' info makes it easy to neglect these terms if you
want to.

On Thu, 25 Jun 2020, 10:16 Brendan McKay, <Brendan.McKay at anu.edu.au> wrote:

> Hi Neil,
>
> Anyone who sees an error on Wikipedia, and knows how to fix it but
> doesn't, is
> responsible for the error.
>
> But in this case there is no error.  Wikipedia follows "reliable
> sources" and in this case
> the reliable source is Golumb's book "Polyominoes".  From a brief look
> it seems that
> Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and
> D.1), so
> Wikipedia is *required* to either also disallow it or to change to a
> different source.
>
> Incidentally, the first comment on A0000105, which is
> "A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the
> moment.
>
> Cheers, Brendan.
>
> On 25/6/20 5:27 pm, Neil Sloane wrote:
> > Everything in the Wikipedia is inaccurate.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From christianperfect at gmail.com  Thu Jun 25 14:25:22 2020
From: christianperfect at gmail.com (Christian Lawson-Perfect)
Date: Thu, 25 Jun 2020 13:25:22 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
 <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
Message-ID: <CAEMHSOhhdCqsKjoKtQ6ah96YmQUvsSxX-26D2BqPcq9osebcrQ@mail.gmail.com>

This feels like the discussion about whether empty matrices exist. See "An
empty exercise" by Carl de Boor -
https://dl.acm.org/doi/10.1145/122272.122273

On Thu, 25 Jun 2020 at 13:15, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr>
wrote:

> Hello everyone,
>
> Thank you Brendan to "bring some water to my mill", and I am sorry to
> "shake the tree up to the top", but a request (actually 3) on OEIS with
> "number of fixed/one-sided/free polyominoes" give results that show this
> question is not definitely clear.
> Moreover, the sequences such as A292357 could be strangely impacted if we
> admit that there is a "zeromino" ; in that case, we should complete the
> first diagonal with a "1" at the beginning (if I follow your hypothesis),
> but what should we put in the corresponding line and row ?? And what could
> mean "a polyomino with height = 0 and width = n" ??
> And moreover again, why should we consider there is ONE zeromino ? Why not
> zero ?
>
> The usual "rule" is (IMO) to exhibit a publication which justify such a
> choice ; till now, I have seen nothing.
>
> Thank you for your attention.
> Best regards,
>
> JLM
>
> ----- Mail original -----
> De: "Brendan McKay" <Brendan.McKay at anu.edu.au>
> ?: "seqfan" <seqfan at list.seqfan.eu>
> Envoy?: Jeudi 25 Juin 2020 11:15:55
> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset
> problem
>
> Hi Neil,
>
> Anyone who sees an error on Wikipedia, and knows how to fix it but
> doesn't, is
> responsible for the error.
>
> But in this case there is no error.  Wikipedia follows "reliable
> sources" and in this case
> the reliable source is Golumb's book "Polyominoes".  From a brief look
> it seems that
> Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and
> D.1), so
> Wikipedia is *required* to either also disallow it or to change to a
> different source.
>
> Incidentally, the first comment on A0000105, which is
> "A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the
> moment.
>
> Cheers, Brendan.
>
> On 25/6/20 5:27 pm, Neil Sloane wrote:
> > Everything in the Wikipedia is inaccurate.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From jean-paul.allouche at imj-prg.fr  Thu Jun 25 19:14:54 2020
From: jean-paul.allouche at imj-prg.fr (jean-paul allouche)
Date: Thu, 25 Jun 2020 19:14:54 +0200
Subject: [seqfan] empty polyomino?
Message-ID: <0e1b8584-0bcf-7011-76eb-d15c81e83ca9@imj-prg.fr>

Hi

If a publication is a justification (I of course agree), open 
/scholar.google.com/,
type "empty polyomino" (with quotes), and contemplate...
An amusing point is that one of the paper says: "where we assume that w is
the contour of a non-empty polyomino" :-)

jean-paul


From fred.lunnon at gmail.com  Thu Jun 25 21:12:53 2020
From: fred.lunnon at gmail.com (Fred Lunnon)
Date: Thu, 25 Jun 2020 20:12:53 +0100
Subject: [seqfan] Re: Polyominoes (A000105, A000988,
 A001168) - offset problem
In-Reply-To: <CAEMHSOhhdCqsKjoKtQ6ah96YmQUvsSxX-26D2BqPcq9osebcrQ@mail.gmail.com>
References: <578447987.10700849.1593004803861.JavaMail.zimbra@unicaen.fr>
 <CAN57Yqt6Fs5Fraf53WLehTqEWys0mD3fg+Q4-Fo7kacdG2v+Nw@mail.gmail.com>
 <2057964585.10863440.1593014353454.JavaMail.zimbra@unicaen.fr>
 <CAAOnSgQvLy+wQV48eWq=xPprOQ7iQPUJxmgsvrPY+Nh0v3Xv-A@mail.gmail.com>
 <98ac4a82-b1bb-1a57-40f7-06334c6bf6b4@anu.edu.au>
 <1791025065.518531.1593087346090.JavaMail.zimbra@unicaen.fr>
 <CAEMHSOhhdCqsKjoKtQ6ah96YmQUvsSxX-26D2BqPcq9osebcrQ@mail.gmail.com>
Message-ID: <CAN57YqufbtJoTZ4qW3ykDgcHGHA3x_PM4cq5B7sHiTU+YTwmZg@mail.gmail.com>

  Or indeed,
    https://en.wikipedia.org/wiki/Field_with_one_element

  Such questions may prove highly nontrivial, and their detailed investigation
reveal subtle disparities between apparently equivalent concepts, as well as
unexpected connections between apparently distinct ones.

  Didactically convenient simplified responses should therefore be avoided
--- but just don't mention  b*n*m*a* c*e*f*c*e*t*  !

WFL


On 6/25/20, Christian Lawson-Perfect <christianperfect at gmail.com> wrote:
> This feels like the discussion about whether empty matrices exist. See "An
> empty exercise" by Carl de Boor -
> https://dl.acm.org/doi/10.1145/122272.122273
>
> On Thu, 25 Jun 2020 at 13:15, Jean-Luc Manguin <jean-luc.manguin at unicaen.fr>
> wrote:
>
>> Hello everyone,
>>
>> Thank you Brendan to "bring some water to my mill", and I am sorry to
>> "shake the tree up to the top", but a request (actually 3) on OEIS with
>> "number of fixed/one-sided/free polyominoes" give results that show this
>> question is not definitely clear.
>> Moreover, the sequences such as A292357 could be strangely impacted if we
>> admit that there is a "zeromino" ; in that case, we should complete the
>> first diagonal with a "1" at the beginning (if I follow your hypothesis),
>> but what should we put in the corresponding line and row ?? And what could
>> mean "a polyomino with height = 0 and width = n" ??
>> And moreover again, why should we consider there is ONE zeromino ? Why not
>> zero ?
>>
>> The usual "rule" is (IMO) to exhibit a publication which justify such a
>> choice ; till now, I have seen nothing.
>>
>> Thank you for your attention.
>> Best regards,
>>
>> JLM
>>
>> ----- Mail original -----
>> De: "Brendan McKay" <Brendan.McKay at anu.edu.au>
>> ?: "seqfan" <seqfan at list.seqfan.eu>
>> Envoy?: Jeudi 25 Juin 2020 11:15:55
>> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset
>> problem
>>
>> Hi Neil,
>>
>> Anyone who sees an error on Wikipedia, and knows how to fix it but
>> doesn't, is
>> responsible for the error.
>>
>> But in this case there is no error.  Wikipedia follows "reliable
>> sources" and in this case
>> the reliable source is Golumb's book "Polyominoes".  From a brief look
>> it seems that
>> Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and
>> D.1), so
>> Wikipedia is *required* to either also disallow it or to change to a
>> different source.
>>
>> Incidentally, the first comment on A0000105, which is
>> "A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the
>> moment.
>>
>> Cheers, Brendan.
>>
>> On 25/6/20 5:27 pm, Neil Sloane wrote:
>> > Everything in the Wikipedia is inaccurate.
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


From tomaszordowski at gmail.com  Wed Jun 24 13:14:22 2020
From: tomaszordowski at gmail.com (Tomasz Ordowski)
Date: Wed, 24 Jun 2020 13:14:22 +0200
Subject: [seqfan] Related to Wolstenholme's theorem
In-Reply-To: <CAF0qcNOzrv_sRqP9Tsw+hhTEQgJ_obJk0EzTW9MS8_4z=2ymfw@mail.gmail.com>
References: <CAF0qcNOzrv_sRqP9Tsw+hhTEQgJ_obJk0EzTW9MS8_4z=2ymfw@mail.gmail.com>
Message-ID: <CAF0qcNP=ocrhjnMswk_0P9kk_9ELay_g8cRCM_yj57hJ+yf_hw@mail.gmail.com>

Dear readers!

Let F(n) = Sum_{k=1..n} 2^{k-1}/k = N(n)/D(n):
1/1, 2/1, 10/3, 16/3, 128/15, 208/15, 2416/105, ...
Theorem: If p > 3 is prime, then p F(p) == 1 (mod p^3);
equivalently p N(p) == D(p) (mod p^4), since p | D(p).
Conjecture: For n > 3, if n N(n) == D(n) (mod n^4),
then n is prime. If so, there are no such pseudoprimes.
Are there the weak pseudoprimes, except n = 49 ?
Composites n such that n N(n) == D(n) (mod n^3):
7^2, 16843^2, 2124679^2, ... To be confirmed.
Primes p such that p N(p) == D(p) (mod p^5)
are 7, 16843, ?  Cf. A088164:
https://oeis.org/A088164
Is p = 2124679 as well?

Best regards,

Thomas Ordowski
_______________________
https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem
     Note that N(n) = A108866(n)/2 : https://oeis.org/A108866
and D(2n+1) = D(2n+2) = A229726(n) : https://oeis.org/A229726