A097496 Subsequence of terms of odd index of the Somos-5 sequence.

1, 1, 2, 5, 37, 274, 6161, 165713, 9434290, 1013908933, 142844426789, 57760865728994, 23510036246274433, 29636604976524724225, 42939094881045167060258, 139127528853472250968193381, 802780326236562434930932841989, 6388127622103180665094420305200242

Offset: 0

Author

Andrew Hone, Aug 24 2004

Keywords

• nonn

Comments

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=(-7/12,-3) and P=(17/12,-1).

Another sequence of points is (2*n - 1)*P on the curve "102a1": y^2 + x * y = x^3 + x^2 - 2 * x where P = [-1, -1]. - Michael Somos, Jul 15 2011

Links

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

Crossrefs

Cf. A006721, A006720.

Programs

Maple

a[0]:=1; a[1]:=1;a[2]:=2;a[3]:=5; for n from 1 to 20 do a[n+3]:=(a[n+2]*a[n]+8*a[n+1]^2)/a[n-1] od;

PARI

{a(n) = sqrtint( denominator( ellpow( ellinit( [1, 1, 0, -2, 0], 1), [-1, -1], 2*n - 1)[1]))}; /* Michael Somos, Jul 15 2011 */

Formula

a(0)=a(1)=1, a(2)=2, a(3)=5; a(n) = (a(n-1)*a(n-3)+8*a(n-2)^2)/a(n-4).

Exact formula: a(n)=D*E^n*sigma(f+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, D=1/sigma(f)=-0.363554228-0.803200610*I, E=sigma(k)*sigma(f)/sigma(f+k)=0.644801269+0.734118205*I, f=integral{infinity}^{-7/12} dx/y = -0.509286773+0.973928783*I, k=integral{17/12}^{infinity} dx/y = 1.018573545 all to 9 d.p.

a(n) = a(1 - n), a(n) = (-8 * a(n-1) * a(n-4) + 57 * a(n-2) * a(n-3)) / a(n-5) for all n in Z. - Michael Somos, Jul 15 2011