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A083527 a(n) is the number of times that sums 1+-4+-9+-16+-...+-n^2 of the first n squares is zero. There are 2^(n-1) choices for the sign patterns. 12
0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 5, 0, 0, 43, 57, 0, 0, 239, 430, 0, 0, 2904, 5419, 0, 0, 27813, 50213, 0, 0, 348082, 649300, 0, 0, 3913496, 7287183, 0, 0, 50030553, 93696497, 0, 0, 611793542, 1161079907, 0, 0, 8009933135, 15176652567, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,12
COMMENTS
The frequency of each possible sum is computed by the Mathematica program without explicitly computing the individual sums.
a(n) is the maximal number of subsets of the first n squares that share the same sum. Cf. A025591, A083309.
a(n)=0 when n==1 or 2 (mod 4).
LINKS
Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..500 (first 240 terms from Alois P. Heinz)
FORMULA
a(n) is half the coefficient of x^0 in the product_{k=1..n} x^(k^2)+x^(k^-2).
a(n) = A158092(n)/2.
a(n) = [x^(n^2)] Product_{k=1..n-1} (x^(k^2) + 1/x^(k^2)). - Ilya Gutkovskiy, Feb 01 2024
EXAMPLE
a(7) = 1 because there is only one sign pattern of the first seven squares that yields zero: 1+4-9+16-25-36+49.
MAPLE
b:= proc(n, i) option remember; local m;
m:= (1+(3+2*i)*i)*i/6;
`if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1) +b(n+i^2, i-1)))
end:
a:= n-> `if`(irem(n-1, 4)<2, 0, b(n^2, n-1)):
seq(a(n), n=1..40); # Alois P. Heinz, Oct 31 2011
MATHEMATICA
d={1, 1}; nMax=60; zeroLst={0}; Do[p=n^2; d=PadLeft[d, Length[d]+p]+PadRight[d, Length[d]+p]; If[1==Mod[Length[d], 2], AppendTo[zeroLst, d[[(Length[d]+1)/2]]], AppendTo[zeroLst, 0]], {n, 2, nMax}]; zeroLst/2
p = 1; t = {}; Do[p = Expand[p(x^(n^2) + x^(-n^2))]; AppendTo[t, Select[p, NumberQ[ # ] &]/2], {n, 51}]; t (* Robert G. Wilson v, Oct 31 2005 *)
PROG
(PARI) a(n)=sum(i=0, 2^(n-1)-1, sum(j=1, n-1, (-1)^bittest(i, j-1)*j^2)==n^2) \\ Charles R Greathouse IV, Nov 05 2012
CROSSREFS
Sequence in context: A263913 A075534 A221361 * A221240 A113038 A082512
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 29 2003
STATUS
approved

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Last modified July 23 08:38 EDT 2024. Contains 374546 sequences. (Running on oeis4.)