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A320769 a(n) where a(n)=-a(-n), a(1)=a(2)=a(3)=a(4)=1, and a(n+2)*a(n-2) = a(n+1)*a(n-1) - c(n)*a(n)^2 where c(3*k)=-2, else c(n)=1.

%I A320769 #13 Jan 05 2025 19:51:41
%S A320769 0,1,1,1,1,3,2,-7,-13,-25,3,-173,-332,1237,2149,12969,-34411,212159,
%T A320769 729350,-5405899,2412231,-129451889,951511591,-6624402137,
%U A320769 -19829335448,484740289833,-2548271136343,27842908929425,-353158277960887,5055074341844027
%N A320769 a(n) where a(n)=-a(-n), a(1)=a(2)=a(3)=a(4)=1, and a(n+2)*a(n-2) = a(n+1)*a(n-1) - c(n)*a(n)^2 where c(3*k)=-2, else c(n)=1.
%C A320769 Let w := (-2)^(1/3), t(n) = w * a(n) if n = 3*k else t(n) = a(n). Then t(n+2)*t(n-2) = t(n+1)*t(n-1) - w*t(n)^2 for all n in Z.
%C A320769 t(n) is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = w, z = 1.
%C A320769 Given elliptic curve 196.a2 E: y^2 = x^3 - x^2 - 2*x + 1 and P = (0,1) the generator, then n*P = ( -w^2*t(n-1)*t(n+1) / t(n)^2, t(2*n) / t(n)^4).
%C A320769 a(n+7)*a(n) = a(n+1)*a(n+6) -2*a(n+3)*a(n+4), a(n+8)*a(n) = 3*a(n+2)*a(n+6) -2*a(n+3)*a(n+5), for all n in Z.
%H A320769 C. Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%H A320769 LMFDB, <a href="http://www.lmfdb.org/EllipticCurve/Q/196/a/2">Elliptic Curve 196.a2 (Cremona label "196a1")</a>.
%F A320769 a(2*n) = a(n)*(c(n-1)*a(n-1)^2*a(n+2) - c(n+1)*a(n+1)^2*a(n-2)) for all n in Z.
%F A320769 c(n) is A061347(n). - _Michael Somos_, Nov 27 2019
%e A320769 G.f. = x + x^2 + x^3 + x^4 + 3*x^5 + 2*x^6 - 7*x^7 - 13*x^8 + ...
%o A320769 (PARI) {a(n) = my(v); if( n==0, 0, n<0, -a(-n), v = vector(n, k, 1); for( k=5, n, v[k] = (v[k-1] * v[k-3] - v[k-2]^2 * [1, 1, -2] [k%3 + 1]) / v[k-4]); v[n])};
%Y A320769 Cf. A061347.
%K A320769 sign
%O A320769 0,6
%A A320769 _Michael Somos_, Oct 20 2018