0,4

The sequence corresponds to the sequence of points Q+nP on the curve y^2=4*x^3-(121/12)*x+845/216, where Q=(-19/12,2) and P=(17/12,-1).

For every 5th order bilinear recurrence of Somos-5 type, b(n+3)*b(n-2)=alpha*b(n+2)*b(n-1)+beta*b(n+1)*b(n) (alpha, beta constant), both the subsequence of even index a(n)=b(2n) and that of odd index a(n)=b(2n+1) satisfy the same 4th order Somos-4 type recurrence a(n+2)*a(n-2)=gamma*a(n+1)*a(n-1)+delta*a(n)^2, where the constant coefficients gamma, delta can be given in terms of alpha, beta and the initial data b(0), b(1), b(2), b(3), b(4).

A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.

a(n) = (a(n-1)*a(n-3)+8*a(n-2)^2)/a(n-4).

Exact formula: a(n)=A*B^n*sigma(c+n*k)/sigma(k)^(n^2) where sigma is the Weierstrass sigma function associated to the elliptic curve y^2=4*x^3-(121/12)*x+845/216, A=1/sigma(c)=0.142427718-1.037985022*I, B=sigma(k)*sigma(c)/sigma(c+k)=0.341936209+0.389300717*I, c=\int_{\infty}^{-19/12} dx/y = 0.163392410+0.973928783*I, k=\int_{17/12}^{\infty} dx/y = 1.018573545 all to 9 d.p.

Cf. A006721, A006720.

Sequence in context: A129114 A136649 A062580 this_sequence A157980 A092148 A091547

Adjacent sequences: A097492 A097493 A097494 this_sequence A097496 A097497 A097498

nonn

Andrew Hone (anwh(AT)kent.ac.uk), Aug 24 2004