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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) T. D. Noe, Extremal Sums of Sequences 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 Cf. A015818, A058498, A063865, A113263, A158092. Sequence in context: A263913 A075534 A221361 * A221240 A113038 A082512 Adjacent sequences: A083524 A083525 A083526 * A083528 A083529 A083530 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.)