[seqfan] Re: Partial sum of Pascal's triangle is a power of 2.

N. J. A. Sloane njas at research.att.com
Wed Nov 24 22:31:43 CET 2010

Robert, Sorry for delay in answering this - I have been
busy with the launch of the new OEIS!

The answer is that Yes, these questions have
been dealt with exhaustively in the work by
several coding theorists and number theorists
(J H van Lint, Aimo Tietavainen, especially)
showing that there are no more perfect codes than
those already known.

I'll add some pertinent references to A171886

 Best regards

>From: Robert Munafo <mrob27 at gmail.com>
>Date: Tue, 26 Oct 2010 02:58:37 -0400
>Subject: Re: Partial sum of Pascal's triangle is a power of 2.

>I didn't hear back from anyone, and then forgot about it -- so it took a few
>months for me to realize I had never submitted the sequence.

>In the meantime, another seqfan reader linked A008949 to the
>as-yet-nonexistent "A171886". That link works now.

>   http://oeis.org/wiki/A171886

>NJAS added the 1977 reference to MacWilliams and Sloane. I do not have
>access to that book. Does anyone know if it proves or disproves the
>proposition that 1+23+253+1771=2048 is the only non-trivial[1] occurrence of
>a power of 2 in sequence A008949? Or is there any other later work that

>- Robert Munafo

>[1] "Trivial": Referring to the example partial sums in
>http://oeis.org/wiki/A008949, examples of "trivial" occurrences of powers of
>2 include the first and last terms of each row, and the "8" and "64" in the
>row "1,8,29,64,99,120,127,128".

>On Sun, May 30, 2010 at 04:36, Robert Munafo <mrob27 at gmail.com> wrote:

>> I want to add this sequence, but I know there must be research more recent
>> than Leech (1964). In particular, I suspect by now someone *must* have
>> proven that the sum 1 + 23 + 23*22/2 + 23*22*21/6 is the only non-trivial
>> case of a partial sum of binomial coefficients that adds up to a power of 2.
>> A008949 is mentioned (not by name of course) in Leech's 1964 paper. The row
>> 23, 0..3 case is significant because it led to the Leech lattice.
>> Neil, I'm CC'ing you because of all your work relating to sphere packing.
>> I've been learning the basics related to the Monster group and the Leech
>> lattice, so of course I ran across this when reading Leech's paper.
>> Note that "A171886" has not been submitted yet, that is an A-number I
>> reserved with the dispenser a few months ago.
>> I am also contemplating adding a comment or two to A008949, like that shown
>> here, but have not done so yet.
>> I need more or better references (Leech 1964 sec. 2.2 talks about the
>> problem but there must be lots of papers about it)
>> %I A171886 [...]

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