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Search: id:A158465
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| A158465 |
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Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0. |
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1
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| 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 16, 18, 0, 0, 32, 100, 0, 0, 424, 510, 0, 0, 2792, 5988, 0, 0
(list; graph; listen)
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OFFSET
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1,16
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COMMENT
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Constant term in the expansion of (x + 1/x)(x^16 + 1/x^16)..(x^n^4 + 1/x^n^4).
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^4) dx
Asymptotic formula:
a(n)=(2^n)*sqrt(18/(pi*n^9))*(1+o(1)) as n-->infty; n=-1 or 0 (mod 4).
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EXAMPLE
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For n=16 the a(16)=2 solutions are
+1+16+81+256-625-1296-2401+4096+6561+10000+14641+20736-28561-38416-50625+65536=0
and the opposite.
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MAPLE
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N:=32: p:=1 a:=[]: for n from 32 to N do p:=expand
(p*(x^(n^4)+x^(-n^4))): a:=[op(a), coeff(p, x, 0)]: od:a;
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CROSSREFS
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Cf. A063865, A158092, A158118, A158380, A019568
Sequence in context: A061848 A120556 A120560 this_sequence A003193 A108474 A120582
Adjacent sequences: A158462 A158463 A158464 this_sequence A158466 A158467 A158468
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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 19 2009
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