A178621 A (1, 2) Somos-4 sequence associated to the elliptic curve E: y^2 + x*y - y = x^3 - x.

1, 1, -2, 5, 13, 24, -229, -365, 7394, -59449, -572233, -7773360, 151071097, -578687351, -87627413666, 1616831255645, 95461592201461, 5420484489890376, -308006807300556157, 18159275356643111275, 3018684797307268036418

Offset: 1

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/(1 - 2x^2/(1 + (5/4)x^2/(1 - (26/25)x^2/(1 + (120/169)x^2/(1 - ... where

2, -5/4, 26/25, -120/169 are the x-coordinates of the multiples of z=(0,0) on E.

Examples

			G.f. = x + x^2 - 2*x^3 + 5*x^4 + 13*x^5 + 24*x^6 - 365*x^7 + ... - _Michael Somos_, Sep 19 2018
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..142 (offset adapted by _Georg Fischer_, Jan 31 2019).

Programs

Magma

I:=[1,1,-2,5]; [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 18 2018

Mathematica

RecurrenceTable[{a[1]==a[2]==1,a[3]==-2,a[4]==5,a[n]==(a[n-1]a[n-3]+ 2a[n-2]^2)/a[n-4]},a[n],{n,30}] (* Harvey P. Dale, Sep 30 2011 *)

PARI

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

PARI

m=30; v=concat([1,1,-2,5], 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 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+3)*a(n-2) = -2*a(n+2)*a(n-1) - 5*a(n+1)*a(n) for all n in Z. - Michael Somos, Sep 19 2018

Extensions

Changed offset to 1 by Michael Somos, Sep 19 2018