A178627 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 + x*y - y = x^3 - x^2 + x and point (0,0).

0, 1, 1, -1, -2, -1, 5, 9, -8, -41, -61, 241, 770, -271, -8649, -27329, 106768, 651521, 740665, -18425809, -107300098, 399122991, 5615422669, 24055184809, -383354254360, -3943757411849, 9276714153611, 498726356978849

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-x^2/(1-2x^2/(1-(1/4)x^2/(1-10x^2/(1-(9/25)x^2/(1-...))))))) where 1,2,-1/4,10,9/25,... are the x-coordinates of the multiples of z=(0,0) on E:y^2+xy-y=x^3-x^2+x.

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

Examples

			G.f. = x + x^2 - x^3 - 2*x^4 - x^5 + 5*x^6 + 9*x^7 - 8*x^8 + ... - _Michael Somos_, Sep 19 2018
		

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.

Programs

Magma

I:=[0, 1, 1, -1, -2]; [n le 5 select I[n] else (Self(n-1)*Self(n-3)+Self(n-2)^2)/Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 06 2014

Mathematica

Join[{0},RecurrenceTable[{a[1]==a[2]==1,a[3]==-1,a[4]==-2,a[n]==(a[n-1]* a[n-3]+ a[n-2]^2)/a[n-4]},a,{n,30}]] (* Harvey P. Dale, Dec 19 2015 *)

PARI

{a(n) = my(E, z); E = ellinit( [1, -1, -1, 1, 0]); z = ellpointtoz( E, [0, 0]); round( ellsigma( E, n*z) / ellsigma( E, z)^(n^2))};

PARI

m=30; v=concat([0,1,1,-1,-2], 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) + a(n-2)^2)/a(n-4), n>4.

a(n) = -a(-n) for all n in Z.

Extensions

Added missing a(0) = 0. - Michael Somos, Aug 06 2014