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

[Robert Munafo]

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
does?

- 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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