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A158092
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Number of solutions to +-1+-2^2+-3^2+-4^2...+-n^2 = 0.
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5
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0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 10, 0, 0, 86, 114, 0, 0, 478, 860, 0, 0, 5808, 10838, 0, 0, 55626, 100426, 0, 0, 696164, 1298600, 0, 0, 7826992, 14574366, 0, 0, 100061106, 187392994, 0, 0, 1223587084, 2322159814, 0, 0, 16019866270, 30353305134, 0, 0, 207366863690, 395899408130, 0, 0, 2773156759854, 5301226883378
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,7
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COMMENTS
| Number of sum partitions of the half of the n-th-square-pyramidal number by distinct square numbers in the range 1 to n^2. Example: a(7)=2 since, squarePyramidal(7)=140 and 70=1+4+16+49=9+25+36. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Oct 20 2010
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FORMULA
| Constant term in the expansion of (x + 1/x)(x^4 + 1/x^4)..(x^n^2 + 1/x^n^2).
a(n)=0 for any n == 1 or 2 (mod 4).
Integral representation:
a(n)=((2^n)/pi)*int_0^pi prod_{k=1}^n cos(x*k^2) dx
Asymptotic formula:
a(n) = (2^n)*sqrt(10/(pi*n^5))*(1+o(1)) as n-->infty; n == -1 or 0 (mod 4).
Equals 2*A083527. - T. D. Noe, Mar 12 2009
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EXAMPLE
| Example: for n=8 the a(8)=2 solutions are: +1-4-9+16-25+36+49-64=0 and -1+4+9-16+25-36-49+64=0
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MAPLE
| Contribution from Pietro Majer (majer(AT)dm.unipi.it), Mar 15 2009: (Start)
N:=60: p:=1: a:=[]: for n from 1 to N do p:=expand(p*(x^(n^2)+x^(-n^2))):
a:=[op(a), coeff(p, x, 0)]: od:a; (End)
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CROSSREFS
| Cf. A063865, A158118, A019568, A083527.
Sequence in context: A063695 A081417 A133388 * A145264 A109042 A128540
Adjacent sequences: A158089 A158090 A158091 * A158093 A158094 A158095
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KEYWORD
| nonn
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AUTHOR
| Pietro Majer (majer(AT)dm.unipi.it), Mar 12 2009
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EXTENSIONS
| a(51)-a(56) from R. H. Hardin (rhhardin(AT)att.net), Mar 12 2009
Edited by N. J. A. Sloane, Sep 15 2009
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