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A332828
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Expansion of (x + x^2 + x^6 - x^7)/(1 - x^2 + x^4 - x^6 + x^8) in powers of x.
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1
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0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1
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OFFSET
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0,1
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COMMENTS
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This is a (-1,1) generalized Somos-4 sequence.
For the elliptic curve y^2 + y = x^3 - x^2, the multiples of the point (0, 0) are (a(n-1)*a(n+1)/a(n)^2, -a(n-1)^2*a(n+2)/a(n)^3).
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LINKS
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FORMULA
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G.f.: (x + x^2 + x^6 - x^7)/(1 - x^2 + x^4 - x^6 + x^8).
a(n) = -a(n+10) = a(5-n) for all n in Z.
a(n) * a(n+4) = -a(n+1) * a(n+3) + a(n+2)^2 for all n in Z.
a(n) * a(n+5) = -a(n+1) * a(n+4) + a(n+2)*a(n+3) for all n in Z.
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EXAMPLE
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G.f. = x + x^2 + x^3 + x^4 + x^6 - x^7 + x^8 - x^9 - x^11 - x^12 + ...
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MATHEMATICA
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a[ n_] := {1, 1, 1, 1, 0, 1, -1, 1, -1, 0}[[Mod[n, 10, 1]]];
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PROG
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(PARI) {a(n) = (-1)^(n\10) * [0, 1, 1, 1, 1, 0, 1, -1, 1, -1][n%10 + 1]};
(PARI) {a(n) = my(E=ellinit([0, -1, 1, 0, 0]), z=ellpointtoz(E, [0, 0])); (-1)^(n\2) * round(ellsigma(E, n*z) / ellsigma(E, z)^n^2)};
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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