A178376 A (-1,-2) Somos-4 sequence associated to the elliptic curve y^2 +y = x^3 +3*x^2 +x.

1, 1, -2, -3, -5, -28, 67, 411, -506, 10855, -74231, -664776, 7518457, -30009367, 1791756790, 22973709333, -762305126477, -10339529833556, -516074985082229, -26431010871217485, 1057255130388472846

Offset: 0

Author

Paul Barry, May 26 2010

Keywords

• easy
• sign

Comments

Hankel transform of the sequence with g.f. 1/(1-x^2/(1+2x^2/(1+(3/4)x^2/(1-(10/9)x^2/(1-...,

where -2,-3/4,10/9,... are the x-coordinates of the multiples of (0,0).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..154
• Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Programs

Magma

I:=[1,1,-2,-3]; [n le 4 select I[n] else (-Self(n-1)*Self(n-3) - 2*Self(n-2)^2)/Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 16 2018

Mathematica

RecurrenceTable[{a[n]==(-a[n-1]*a[n-3] -2*a[n-2]^2)/a[n-4], a[0] == 1, a[1] == 1, a[2] == -2, a[3] == -3}, a, {n, 0, 30}] (* G. C. Greubel, Sep 16 2018 *)

PARI

m=30; v=concat([1,1,-2,-3], vector(m-4)); for(n=5, m, v[n] = ( -v[n-1]*v[n-3] - 2*v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Sep 16 2018

Formula

a(n) = (-a(n-1)*a(n-3) - 2*a(n-2)^2)/a(n-4), n>3.