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

Robert Munafo mrob27 at gmail.com
Sun May 30 10:36:19 CEST 2010

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
%S A171886
%T A171886
%U A171886
%N A171886 Numbers n such that A008949(n) is a power of 2.
%C A171886 Partial sums of binomial coefficients were considered in section
2.2 of the 1964 paper by Leech. The presence of the number 279 corresponds
to the existence of the Leech lattice.
%C A171886 In general, A000217(n+1)+i-1 is in this sequence IFF the first i
items in row n of Pascal's triangle add up to a power of 2.
%C A171886 Almost all members of this sequence are "trivial" terms of four
types: A000217(i); A000217(i)+1, A000217(i)+i, and A000217(2i+1)+i for all
integers i. 279 is the sole non-trivial term.
%E A171886 17 is in the sequence because A008949(17)=16, which in turn is
because the first 3 elements of row 5 of Pascal's triangle, 1+5+10, add up
to 16.
%E A171886 279 is in the sequence because the first 4 elements of row 24 of
Pascal's triangle add up to 2^11: 1+23+253+1771=2048.
%D A171886 John Leech, ``Some Sphere Packings in Higher Space'', Can. J.
Math., 16 (1964), page 669.
%H A171886 John Leech, <a href="
http://www.cms.math.ca/cjm/v16/cjm1964v16.0657-0682.pdf">Some Sphere
Packings in Higher Space</a> (PDF available from the publisher).
%A A171886 Robert Munafo, May 30 2010

%C A008949 Row n, partial sum r gives the number of vertices within distance
r (measured along the edges) of an n-dimensional unit cube, (i.e. the number
of vertices on the hypercube graph Q_n whose distance from a reference
vertex is <= r)

 Robert Munafo  --  mrob.com

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