A178417 A (-1,1) Somos-4 sequence associated to the elliptic curve y^2 + x*y + y = x^3 + x^2 + x.

1, 1, 1, 4, -3, 19, -67, 40, -1243, -4299, -25627, -334324, 627929, -29742841, 372632409, -1946165680, 128948361769, 1488182579081, 52394610324649, 2333568937567764, -5642424912729707, 3857844273728205019

Offset: 1

Author

Paul Barry, May 27 2010

Keywords

• easy
• sign

Comments

Hankel transform of the sequence with g.f. 1/(1-x^2/(1-x^2/(1-4x^2/(1+(3/16)x^2/(1-(76/9)x^2/(1-.... where 1,4,-3/16,76/9,... are the x-coordinates of the multiples of (0,0).

Examples

			G.f. = x + x^2 + x^3 + 4*x^4 - 3*x^5 + 19*x^6 - 67*x^7 + ... - _Michael Somos_, Sep 17 2018
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..156 (offset adapted by _Georg Fischer_, Jan 31 2019).

Programs

Magma

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

PARI

m=30; v=concat([1,1,1,4], vector(m-4)); for(n=5, m, v[n] = ( -v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Sep 16 2018

Formula

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

a(n) = -(-1)^n*a(-n) for all n in Z. - Michael Somos, Sep 17 2018

Extensions

Changed offset to 1 by Michael Somos, Sep 17 2018