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A083527
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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.
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6
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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, 103683431845
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OFFSET
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1,12
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COMMENTS
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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).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..240
T. D. Noe, Extremal Sums of Sequences
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FORMULA
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a(n) is half the coefficient of x^0 in the product_{k=1..n} x^(k^2)+x^(k^-2).
Equals A158092/2.
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 (from Robert G. Wilson v, Oct 31 2005)
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PROG
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(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
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CROSSREFS
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Cf. A015818, A058498, A063865, A113263, A158092.
Sequence in context: A019178 A075534 A221361 * A221240 A113038 A082512
Adjacent sequences: A083524 A083525 A083526 * A083528 A083529 A083530
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe, Apr 29 2003
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STATUS
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approved
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