A178624 A (1,3) Somos-4 sequence associated to the elliptic curve E: y^2 + 2*x*y - y = x^3 - x.

1, 1, -3, 11, 38, 249, -2357, 8767, 496035, -3769372, -299154043, -12064147359, 632926474117, -65604679199921, -6662962874355342, -720710377683595651, 285131375126739646739, 5206174703484724719135

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^2/(1-3x^2/(1+(11/9)x^2/(1-(114/121)x^2/(1+(2739/1444)x^2/(1-... where 3,-11/9,141/121,-2739/1444... are the x-coordinates of the multiples of z=(0,0) on E:y^2+2xy-y=x^3-x.

Examples

			G.f. = x + x^2 - 3*x^3 + 11*x^4 + 38*x^5 + 249*x^6 + ... - _Michael Somos_, Sep 17 2018
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..118 (offset adapted by _Georg Fischer_, Jan 31 2019)

Programs

Magma

I:=[1,1,-3,11]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + 3*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] +3*a[n-2]^2)/a[n-4], a[0] == 1, a[1] == 1, a[2] == -3, a[3] == 11}, a, {n, 0, 30}] (* G. C. Greubel, Sep 16 2018 *)

PARI

a(n)=local(E,z);E=ellinit([2,0,-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,-3,11], vector(m-4)); for(n=5, m, v[n] = ( v[n-1]*v[n-3] + 3*v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Sep 16 2018

Formula

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

a(n) = -a(-n) for all n in Z. - Michael Somos, Sep 17 2018

Extensions

Corrected by Paul Barry, Jun 01 2010

Offset changed to 1 by Michael Somos, Sep 17 2018