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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.

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 December 9 17:36 EST 2022. Contains 358702 sequences. (Running on oeis4.)