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.

0, 1, 1, 1, 1, 3, 2, -7, -13, -25, 3, -173, -332, 1237, 2149, 12969, -34411, 212159, 729350, -5405899, 2412231, -129451889, 951511591, -6624402137, -19829335448, 484740289833, -2548271136343, 27842908929425, -353158277960887, 5055074341844027

Offset: 0

Author

Michael Somos, Oct 20 2018

Keywords

• sign

Comments

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.

t(n) is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = w, z = 1.

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).

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.

Examples

			G.f. = x + x^2 + x^3 + x^4 + 3*x^5 + 2*x^6 - 7*x^7 - 13*x^8 + ...
		

Links

• C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
• LMFDB, Elliptic Curve 196.a2 (Cremona label "196a1").

Crossrefs

Cf. A061347.

Programs

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])};

Formula

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.

c(n) is A061347(n). - Michael Somos, Nov 27 2019