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

Author

Paul Barry, May 31 2010

Keywords

• easy
• sign

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

Examples

			G.f. = x + x^2 + 2*x^3 + x^4 - 7*x^5 - 16*x^6 - 57*x^7 - ... - _Michael Somos_, Oct 22 2024
		

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.

Crossrefs

Cf. A242107.

Programs

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

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 *)
a[ n_] := Which[n == 0, 0, n < 0, -a[-n], n < 5, {1, 1, 2, 1}[[n]], True, a[n] = (a[n-1]*a[n-3] - 2*a[n-2]*a[n-2])/a[n-4]]; (* Michael Somos, Oct 22 2024 *)

PARI

{a(n) = my(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

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

a(n) = A242107(2*n) for all n in Z. - Michael Somos, Oct 22 2024

Extensions

Added missing a(0)=0.

More terms from Vincenzo Librandi, Aug 07 2014