Generalized Go Sequence

Mon May 12 15:16:43 CEST 2008

This is the same sequence
starting with 1:

1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23,35,29,32,16,8,4,2,1,46,23

and corresponding %N:

%N A000001 Rule : If b(n-1) is divisible by two
then b(n) = b(n-1)/2.
If b(n-1) isn't divisible by two then b(n) =
k-(b(n-1)+1)/2. b(0)=1. k=47.

best, zak

--- "N. J. A. Sloane" <njas at research.att.com> wrote:

> Yasutoshi
> 
> Thanks for proposing this sequence:
> 
>     %I A000001
>     %S A000001
>
1000,500,250,125,-16,-8,-4,-2,-1,47,23,35,29,32,16,8,4,2,1,46,23,....
>     %N A000001 Rule : If b(n-1) is divisible by two
> then b(n) = b(n-1)/2.
>     If b(n-1) isn't divisible by two then b(n) =
> k-(b(n-1)+1)/2. b(0)=1000. k=47.
>     %C A000001 For all integers i,j If k=i, b(0)=j
> then b(n) becomes periodic.             
>     %Y A000001 A000002
>     %K A000001 none
>     %O A000001 1,1
>     %A A000001 Yasutoshi Kohmoto  
> zbi74583 at boat.zero.ad.jp
> 
> But I do not like the fact that it begins at 1000.
> Why 1000?
> 
> There iis a rule that sequences in the OEIS should
> not depend
> on a large but arbitrary parameter.
> 
> 
>  Best regards
>  			 Neil
> 
>  Neil J. A. Sloane
>  AT&T Shannon Labs, Room C233, 
>  180 Park Avenue, Florham Park, NJ 07932-0971
>  Email: njas at research.att.com
>  Office: 973 360 8415; fax: 973 360 8178
>  Home page: http://www.research.att.com/~njas/
> 

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Hi everybody,

* N. J. A. Sloane <njas at research.att.com> [May 10. 2008 22:07]:
> Joerg,
> 
> You are right, they were produced by Simon Plouffe
> in his thesis. 
> 
> And Yes, some are trivially true, and some are obviously false.
> But what justified this work - in my opinion - were
> the ones in between, where the g.f. /might/ tell us
> something new.

Yes, agreed.

An automated production of OGFs is definitely fine.

But there are sequences (not in the class indicated below)
where the OGF very likely just interpolates the given terms.
I'd be myself very unhappy if my name would stand next to
an OGF for the Ramsey numbers with 'conjectured by Joerg Arndt'.

There are nontrivial cases where the OGF have a chance to be
actually true.  However, I'd add the OGF only if the
guess-ogf routine produces the OGF for the seq shortened
by a few terms and predicts the remaining terms correctly.
My impression is that this has not been done.

> 
> In 1995 we used some of these in the EIS book
> - but only a few.
> 
> Then last year, Simon noticed that people were starting to add 
> g.f.'s for many sequences, g.f.'s which he had found
> in his thesis.  In order to give him credit, I added 
> most of the other g.f.s from his thesis,
> marking them ALL as conjectures.  Because that is what they are.

For a great number of seqs (e.g. of the form a(n)=some_poly(n),
and exponential sums) we must have a rational OGF, that's
why I find the term 'conjecture' quite a stretch.

I included some material regarding linear homogeneous recurrences
in the fxtbook because people keep rediscovering that old stuff.
I might add a section about recovering the OGF form closed
forms if time allows.

> 
> I am now slowly making changes to these g.f.s.
> If the g.f. is obviously correct, I remove
> the words "Conjected by"
> 
> If it is wrong, I say "The g.f. conjectured by Simon Plouffe
> in his thsis is incorrect."
> 
> The remaining cases will remain as "Conjectured by ..."
> 
> I am glad you have offered to help.  If you send me a bigger
> list of correct g.f.s and incorrect g.f.s,
> I will make the necessary changes.

The issue is that the OGFs will produce the values of the seq
unless the seq has been extended.

E.g. for 
http://www.research.att.com/~njas/sequences/A005045
the 35 given terms are predicted correctly.
Given that the OGF has total degree (deg(num)+deg(den)) 19 
there is a chance of it being correct.

> 
> As to whether this project was worthwhile, I believe it was.
> In the correct cases, Simon gets credit.

Of course!

> In the open cases, this in interesting material,
> and anyone working on one of these sequences will be happy
> to have a g.f. to test.

One might use wording as
"The OGF ... produces the terms given", i.e. avoiding
the term "conjecture", unless Simon really conjectured!

> 
> In the incorrect cases, it is also important to 
> publish the fact that the g.f. in his thesis is wrong.
> For this is misleading material that has been
> published, and so it is important to publish
> the fact that it is wrong!
> 

Again agreed!

> 
> Best regards
> 
> Neil Sloane
> 
> 
>  Neil J. A. Sloane
>  AT&T Shannon Labs, Room C233, 
>  180 Park Avenue, Florham Park, NJ 07932-0971
>  Email: njas at research.att.com
>  Office: 973 360 8415; fax: 973 360 8178
>  Home page: http://www.research.att.com/~njas/

cheers,   jj

* N. J. A. Sloane <njas at research.att.com> [May 10. 2008 22:07]:
> PS
> 
> Joerg,  I think you were a bit harsh.  Here is your list:
> 
> > Whenever the 'conjecture' is not trivially true I suspect
> > it is just wrong, e.g. one might want to check
> > A006172 (and add a definition!)
> > A006186
> > A006672
> >  (wow! OGF for Ramsey numbers -(4+8*z+14*z**2+z**3+5*z**4)/(-1-z-z**2+4*z**3)
> >   Fields medal formula!)
> > A005045
> > A005103
> > A005308 (definitely WRONG)
> > A005309
> > A005310
> > A004129
> > A004138
> > (and, sadly, very many more)
> > What a mess.
> 
> Me:  I agree that many of these are wrong. But are
> you so sure A005045 is wrong?

I cannot test (but hopefully Brendan McKay will be able to help,
if not, I'll do a computation).

The OGF is also obtained with the last 7 terms dropped,

> (Of course you may be more familiar with Elizabeth Morgan's
> work than I am.)

Assuming me to know more than you about any kind of math is,
uhm... likely to be wrong  8-)

> And even A004129 and A004138 cannot be easily dismissed.

A004129: total deg of OGF == number of terms given!
Ralf's ggf() does not produce a OGF.
(same for A005103)

A004138:
OGF also with last 8 terms dropped, so there is hope.

> That's three out of 10 conjectures that are at least worth thinking about.
> 
> IMHO one pearl justifies sifting through a lot of oysters!

Yes!

> Even if the work is messy.
> 
> Neil
> 

cheers,   jj

P.S.: you are too lenient!

What is the significance of 47?
Also the seq appears to be trivially periodic(?)

What does the name refer to?

* zak seidov <zakseidov at yahoo.com> [May 12. 2008 16:28]:
> This is the same sequence
> starting with 1:
> 
>
1,46,23,35,29,32,16,8,4,2,
1,46,23,35,29,32,16,8,4,2,
1,46,23,35,29,32,16,8,4,2,
1,46,23,35,29,32,16,8,4,2,
etc.

> 
> and corresponding %N:
> 
> %N A000001 Rule : If b(n-1) is divisible by two
> then b(n) = b(n-1)/2.
> If b(n-1) isn't divisible by two then b(n) =
> k-(b(n-1)+1)/2. b(0)=1. k=47.
> 
> best, zak
> 
> [...]

A038048 == (n-1)! * sum {d\n} d
A110375 == n! * sum {d\n} 1/d