%I #37 Feb 03 2024 10:15:17

%S 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,

%T 27813,50213,0,0,348082,649300,0,0,3913496,7287183,0,0,50030553,

%U 93696497,0,0,611793542,1161079907,0,0,8009933135,15176652567,0,0

%N 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.

%C The frequency of each possible sum is computed by the Mathematica program without explicitly computing the individual sums.

%C a(n) is the maximal number of subsets of the first n squares that share the same sum. Cf. A025591, A083309.

%C a(n)=0 when n==1 or 2 (mod 4).

%H Alois P. Heinz and Ray Chandler, <a href="/A083527/b083527.txt">Table of n, a(n) for n = 1..500</a> (first 240 terms from Alois P. Heinz)

%H T. D. Noe, <a href="http://www.sspectra.com/math/Sums.html">Extremal Sums of Sequences</a>

%F a(n) is half the coefficient of x^0 in the product_{k=1..n} x^(k^2)+x^(k^-2).

%F a(n) = A158092(n)/2.

%F a(n) = [x^(n^2)] Product_{k=1..n-1} (x^(k^2) + 1/x^(k^2)). - _Ilya Gutkovskiy_, Feb 01 2024

%e 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.

%p b:= proc(n, i) option remember; local m;

%p m:= (1+(3+2*i)*i)*i/6;

%p `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1) +b(n+i^2, i-1)))

%p end:

%p a:= n-> `if`(irem(n-1, 4)<2, 0, b(n^2, n-1)):

%p seq(a(n), n=1..40); # _Alois P. Heinz_, Oct 31 2011

%t 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

%t 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 *)

%o (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

%Y Cf. A015818, A058498, A063865, A113263, A158092.

%K nonn

%O 1,12

%A _T. D. Noe_, Apr 29 2003