A178622 A (1, -2) Somos-4 sequence associated to the elliptic curve E: y^2 - 3*x*y - y = x^3 - x.

0, 1, 1, 2, 1, -7, -16, -57, -113, 670, 3983, 23647, 140576, -833503, -14871471, -147165662, -2273917871, 11396432249, 808162720720, 14252325989831, 503020937289311, 23268424032702, -625775582778294689, -22086170583356766977, -1557994930804790259136, -27620103680757212617727, 6783061219100782906098017, 547569584492952570186575810

Offset

0,4

Comments

a(n) is (-1)^C(n,2) times the Hankel transform of the sequence with g.f. 1/(1-x^2/(1+2x^2/(1+(1/4)x^2/(1-14x^2/(1-(16/49)x^2/(1-... where 0/1, -2/1, -1/4, 14/1, 16/49, ... are the x-coordinates of the multiples of z=(0, 0) on E.

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 2, z = 1. - Michael Somos, Aug 06 2014

Links

G. C. Greubel, Table of n, a(n) for n = 0..155

Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

Formula

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

a(n) = -a(-n), a(n+5)*a(n) = 2*a(n+4)*a(n+1) - a(n+3)*a(n+2) for all n in Z. - Michael Somos, Aug 06 2014

Example

G.f. =

Mathematica

nxt[{a_, b_, c_, d_}]:={b, c, d, (d*b-2c^2)/a}; Join[{0}, Transpose[ NestList[ nxt, {1, 1, 2, 1}, 30]][[1]]] (* Harvey P. Dale, Aug 19 2015 *)
Join[{0}, RecurrenceTable[{a[n] == (a[n-1]*a[n-3] -2*a[n-2]^2)/a[n - 4], a[1] == 1, a[2] == 1, a[3] == 2, a[4] == 1}, a, {n, 1, 30}]] (* G. C. Greubel, Sep 18 2018 *)

Prog

(PARI) a(n)=local(E, z); E=ellinit([3, 0, 1, -1, 0]); z=ellpointtoz(E, [0, 0]); -(-1)^n*round(ellsigma(E, n*z)/ellsigma(E, z)^(n^2))
(PARI) m=30; v=concat([0, 1, 1, 2, 1], vector(m-5)); for(n=6, m, v[n] = ( v[n-1]*v[n-3] - 2*v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Sep 18 2018
(Magma) I:=[0, 1, 1, 2, 1]; [n le 5 select I[n] else (Self(n-1)*Self(n-3)-2*Self(n-2)^2)/Self(n-4): n in [1..30]]; // Vincenzo Librandi, Aug 07 2014

Crossrefs

Sequence in context: A113042 A305881 A184346 * A013070 A012888 A012893

Adjacent sequences: A178619 A178620 A178621 * A178623 A178624 A178625

Keyword

easy,sign

Author

Paul Barry, May 31 2010

Extensions

Added missing a(0)=0.

More terms from Vincenzo Librandi, Aug 07 2014

Status

approved