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Search: id:A158118
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| A158118 |
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Number of solutions of +-1+-2^3+-3^3..+-n^3=0. |
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+0
5
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| 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 2, 0, 0, 4, 124, 0, 0, 536, 712, 0, 0, 4574, 2260, 0, 0, 10634, 73758, 0, 0, 406032, 638830, 0, 0, 4249160, 3263500, 0, 0, 21907736, 82561050, 0, 0, 485798436, 945916970, 0, 0, 5968541478, 6839493576, 0, 0
(list; graph; listen)
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OFFSET
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1,12
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COMMENT
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Constant term in the expansion of (x + 1/x)(x^8 + 1/x^8)..(x^n^3 + 1/x^n^3).
a(n)=0 for any n=1 (mod4) or n=2 (mod4).
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FORMULA
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Integral representation:
a(n)=((2^n)/pi)*int_0^pi prod_{k=1}^n cos(x*k^3) dx
Asymptotic formula:
a(n)=(2^n)*sqrt(14/(pi*n^7))*(1+o(1)) as n-->infty; n=-1 or 0 (mod 4).
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EXAMPLE
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Example: For n=12 the a(12)=2 solutions are: +1+8-27+64-125-216-343+512+729-1000-1331+1728=0 -1-8+27-64+125+216+343-512-729+1000+1331-1728=0
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MAPLE
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N:=60: p:=1: a:=[]: for n from 1 to N do p:=expand(p*( x^(n^3) + x^(-n^3) )): a:=[op(a), coeff(p, x, 0)]: od:a;
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CROSSREFS
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A063865
A158092, A019568 [From Pietro Majer (majer(AT)dm.unipi.it), Mar 15 2009]
Adjacent sequences: A158115 A158116 A158117 this_sequence A158119 A158120 A158121
Sequence in context: A106235 A118965 A121552 this_sequence A147592 A108885 A072740
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KEYWORD
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nonn
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AUTHOR
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Pietro Majer (majer(AT)dm.unipi.it), Mar 12 2009
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