[seqfan] Re: parity of binomials and some carryless products A175669 - A175672
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) +
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)
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
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.)
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:
>> 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:
> 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
> (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.
>> Richard Mathar
More information about the SeqFan