A178418 A (-1, 2) Somos-4 sequence associated to the elliptic curve y^2 + 2*x*y + y = x^3 + x^2 + x.

1, 1, 2, 9, -1, 164, -737, 5895, -119558, -113489, -39697759, -800750664, -25602199327, -2344309254991, 14264330936074, -14182654502256615, 1282764246790221919, -163799892405003723284, 67851128001519788451263

Offset: 1

Author

Paul Barry, May 27 2010

Keywords

• easy
• sign

Comments

Hankel transform of the sequence with g.f. 1/(1-x^2/(1-2x^2/(1-(9/4)x^2/(1+(2/81)x^2/(1-1476x^2/(1-.... where 0/1, 2/1, 9/4, -2/81, 1476/1,... are the x-coordinates of the multiples of (0, 0).

Examples

			G.f. = x + x^2 + 2*x^3 + 9*x^4 - x^5 + 164*x^6 - 737*x^7 + ... - _Michael Somos_, Sep 19 2018
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..125 (offset adapted by _Georg Fischer_, Jan 31 2019)

Programs

Magma

I:=[1,1,2,9]; [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]==9,a[n]==(-a[n-1]a[n-3]+ 2a[n-2]^2)/a[n-4]},a[n],{n,20}] (* Harvey P. Dale, Sep 20 2011 *)

PARI

m=30; v=concat([1,1,2,9], 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) = -(-1)^n * a(-n), a(n+3)*a(n-2) = -a(n+2)*a(n-1) + 9*a(n+1)*a(n) for all n in Z. - Michael Somos, Sep 19 2018

Extensions

Offset changed to 1 by Michael Somos, Sep 19 2018