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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.

%I A083527 #37 Feb 03 2024 10:15:17
%S A083527 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 A083527 27813,50213,0,0,348082,649300,0,0,3913496,7287183,0,0,50030553,
%U A083527 93696497,0,0,611793542,1161079907,0,0,8009933135,15176652567,0,0
%N 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.
%C A083527 The frequency of each possible sum is computed by the Mathematica program without explicitly computing the individual sums.
%C A083527 a(n) is the maximal number of subsets of the first n squares that share the same sum. Cf. A025591, A083309.
%C A083527 a(n)=0 when n==1 or 2 (mod 4).
%H A083527 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 A083527 T. D. Noe, <a href="http://www.sspectra.com/math/Sums.html">Extremal Sums of Sequences</a>
%F A083527 a(n) is half the coefficient of x^0 in the product_{k=1..n} x^(k^2)+x^(k^-2).
%F A083527 a(n) = A158092(n)/2.
%F A083527 a(n) = [x^(n^2)] Product_{k=1..n-1} (x^(k^2) + 1/x^(k^2)). - _Ilya Gutkovskiy_, Feb 01 2024
%e A083527 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 A083527 b:= proc(n, i) option remember; local m;
%p A083527       m:= (1+(3+2*i)*i)*i/6;
%p A083527       `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1) +b(n+i^2, i-1)))
%p A083527     end:
%p A083527 a:= n-> `if`(irem(n-1, 4)<2, 0, b(n^2, n-1)):
%p A083527 seq(a(n), n=1..40);  # _Alois P. Heinz_, Oct 31 2011
%t A083527 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 A083527 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 A083527 (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 A083527 Cf. A015818, A058498, A063865, A113263, A158092.
%K A083527 nonn
%O A083527 1,12
%A A083527 _T. D. Noe_, Apr 29 2003