OFFSET
0,6
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.
LINKS
C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
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
EXAMPLE
G.f. = x + x^2 + x^3 + x^4 + 3*x^5 + 2*x^6 - 7*x^7 - 13*x^8 + ...
PROG
(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])};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 20 2018
STATUS
approved