[seqfan] Re: parity of binomials and some carryless products A175669 - A175672
Antti Karttunen
antti.karttunen at gmail.com
Thu Aug 12 19:12:31 CEST 2010
Well, it was already there, mentioned by Franklin Adams-Watters:
A048715 Binary expansion matches ((0)*001)*(0*); or,
Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) +
f(n-3). +20
10
0, 1, 2, 4, 8, 9, 16, 17, 18, 32, 33, 34, 36, 64, 65, 66, 68, 72, 73,
128, 129, 130, 132, 136, 137, 144, 145, 146, 256, 257, 258, 260, 264,
265, 272, 273, 274, 288, 289, 290, 292, 512, 513, 514, 516, 520, 521,
528 (list; graph; listen)
OFFSET
No more than one 1-bit in each bit triplet. All terms satisfy A048727(n) = 7*n.
It appears that n is in the sequence if and only if C(7n,n) is odd
(cf. A003714). - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 09
2003
The conjecture by Benoit is correct. This is easily proved using the
well-known result that the multiplicity
with which a prime p divides C(n+m,n) is the number of carries when
adding n+m in base p.
[From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 06 2009]
(It's also hot and humid here... Sorry.)
Yours,
Antti
On Thu, Aug 12, 2010 at 8:02 PM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> On Wed, Aug 11, 2010 at 9:54 PM, <seqfan-request at list.seqfan.eu> wrote:
>> Message: 14
>> Date: Wed, 11 Aug 2010 20:25:16 +0200
>> From: Richard Mathar <mathar at strw.leidenuniv.nl>
>> Subject: [seqfan] parity of binomials and some carryless products,
>> A175669 - A175672
>> To: seqfan at seqfan.eu
>> Message-ID: <201008111825.o7BIPGvG019644 at krypton.strw.leidenuniv.nl>
>> Content-Type: text/plain; charset=us-ascii
>>
>>
>> These 8 sequences appear highly correlated in pairs:
>>
>> http://oeis.org/classic/?q=id:A175672|id:A115845
>> 0,1,2,3,4,5,6,7,8,10,12,14,16,17,20,21,24,28,32,33,34,35,40,42,48,49,56,64,65,66,67,68,69,70,71,80,81,84,85,96,97,98,99,112,113,128,129,130,131,132,133,134,135,136,138,140,142,160,161,162,163,168,170,192,193,
>>
>> 0,1,2,3,4,5,6,7,8,10,12,14,16,17,20,21,24,28,32,33,34,35,40,42,48,49,56,64,65,66,67,68,69,70,71,80,81,84,85,96,97,98,99,112,113,128,129,130,131,132,133,134,135,136,138,140,142,160,161,162,163,168,170,192,
>>
>> http://oeis.org/classic/?q=id:A175671|id:A048715
>> 0,1,2,4,8,9,16,17,18,32,33,34,36,64,65,66,68,72,73,128,129,130,132,136,137,144,145,146,256,257,258,260,264,265,272,273,274,288,289,290,292,512,513,514,516,520,521,528,529,530,544,545,546,548,576,577,578,580,
>>
>> 0,1,2,4,8,9,16,17,18,32,33,34,36,64,65,66,68,72,73,128,129,130,132,136,137,144,145,146,256,257,258,260,264,265,272,273,274,288,289,290,292,512,513,514,516,520,521,528,
>>
>>
>> http://oeis.org/classic/?q=id:A175670|id:A048716
>> 0,1,2,3,4,6,8,9,12,16,17,18,19,24,25,32,33,34,35,36,38,48,49,50,51,64,65,66,67,68,70,72,73,76,96,97,98,99,100,102,128,129,130,131,132,134,136,137,140,144,145,146,147,152,153,192,193,194,195,196,198,200,201,
>>
>> 0,1,2,3,4,6,8,9,12,16,17,18,19,24,25,32,33,34,35,36,38,48,49,50,51,64,65,66,67,68,70,72,73,76,96,97,98,99,100,102,128,129,130,131,132,134,136,137,140,144,145,146,147,
>>
>> http://oeis.org/classic/?q=id:A175669|id:A003714
>> 0,1,2,4,5,8,9,10,16,17,18,20,21,32,33,34,36,37,40,41,42,64,65,66,68,69,72,73,74,80,81,82,84,85,128,129,130,132,133,136,137,138,144,145,146,148,149,160,161,162,164,165,168,169,170,256,257,258,260,261,264,265,
>>
>> 0,1,2,4,5,8,9,10,16,17,18,20,21,32,33,34,36,37,40,41,42,64,65,66,68,69,72,73,74,80,81,82,84,85,128,129,130,132,133,136,137,138,144,145,146,148,149,160,161,162,164,165,168,169,170,256,257,258,260,
>>
>> They all deal with parities of binomials C(k*n,n) on one hand
>> and the Finnish way of multiplication in GF(2)[x] on the other.
>
> The starving bears ate our carry-bits, so we have to do without...
>
>> It would
>> be nice to have some explanations on the mutual correlations
>> and --from the purely administrative point of view-- comments inserted
>> when they start to differ.
>
> Also, regarding the recent speculation about running all new
> submissions through Superseeker:
> it would be nice even if it did start with just one transformation:
> the identity,
> to detect the obvious duplicate-candidates!
>
> BTW, I check every sequence I submit with the python script available from here:
> http://oeis.org/w/images/a/a0/User_files-Antti_Karttunen-oeischek_py.txt
> it creates a similarly named local HTML-file, where the %S-lines
> (after two initial terms)
> are search-links to OEIS. (Actually the subsequence to be searched
> from OEIS should
> be even shorter.)
>
> My hunch is that it shouldn't be too hard to prove that the sequences
> mentioned above
> are identical. If one considers the parity of binomial coefficient of C(kn,n),
> it's the same thing as a bit n on row kn in
> http://oeis.org/classic/table?a=47999&fmt=312
> (and each row is obtained from the previous row with a simple shifted
> xor, see http://oeis.org/classic/A001317 )
>
> But sorry, too tired to do it by myself now.
>
> Cheers,
>
> Antti
>
>>
>> Richard Mathar
>>
>
More information about the SeqFan
mailing list