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%I #9 May 13 2022 05:14:02
%S 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,
%T -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,
%U -1,1,-1,1,0,1,1,1,1,0,1,-1,1,-1,0,-1,-1,-1,-1,0,-1
%N Expansion of (x + x^2 + x^6 - x^7)/(1 - x^2 + x^4 - x^6 + x^8) in powers of x.
%C This is a (-1,1) generalized Somos-4 sequence.
%C 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).
%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/11/a/3">Elliptic Curve with LMFDB label 11.a3 (Cremona label 11a3)</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,-1,0,1,0,-1).
%F G.f.: (x + x^2 + x^6 - x^7)/(1 - x^2 + x^4 - x^6 + x^8).
%F a(n) = -a(n+10) = a(5-n) for all n in Z.
%F a(n) * a(n+4) = -a(n+1) * a(n+3) + a(n+2)^2 for all n in Z.
%F a(n) * a(n+5) = -a(n+1) * a(n+4) + a(n+2)*a(n+3) for all n in Z.
%F a(2*n) = A099443(n-1), a(2*n+1) = A099443(n+2) for all n in Z.
%e G.f. = x + x^2 + x^3 + x^4 + x^6 - x^7 + x^8 - x^9 - x^11 - x^12 + ...
%t a[ n_] := {1, 1, 1, 1, 0, 1, -1, 1, -1, 0}[[Mod[n, 10, 1]]];
%o (PARI) {a(n) = (-1)^(n\10) * [0, 1, 1, 1, 1, 0, 1, -1, 1, -1][n%10 + 1]};
%o (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)};
%Y Cf. A006720, A099443.
%K sign,easy
%O 0,1
%A _Michael Somos_, Feb 25 2020
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