A178079 A (1,-1) Somos-4 sequence.

0, 1, 1, 1, 2, 1, -3, -7, -8, -25, -37, 47, 318, 559, 2023, 7039, -496, -90431, -314775, -1139599, -8007614, -13512079, 154788437, 1247862041, 5097732072, 56844671623, 290379801907, -1403230649825, -32188159859842, -199066111517153

Offset: 0

Author

Paul Barry, May 19 2010

Keywords

• easy
• sign

Comments

Hankel transform of A178078 is a(n+2).

Apparently a(n) = -A174400(n+1). - R. J. Mathar, Dec 10 2010

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

Associated with elliptic curve "61a1" y^2 + x*y = x^3 - 2*x + 1 and point (1, -1). - Michael Somos, Sep 27 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..150
• Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
• Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
• LMFDB, Elliptic Curve 61.a1 (Cremona label 61a1)

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..30]]; // Vincenzo Librandi, Aug 07 2014

Mathematica

Join[{0}, RecurrenceTable[{a[n] == (a[n - 1]*a[n - 3] - a[n - 2]^2)/a[n - 4], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 2}, a, {n, 1, 50}]] (* G. C. Greubel, Sep 22 2018 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,(d*b-c^2)/a}; Join[{0},NestList[nxt,{1,1,1,2},30][[;;,1]]] (* Harvey P. Dale, Sep 27 2023 *)

PARI

m=50; v=concat([1,1,1,2], vector(m-4)); for(n=5, m, v[n] = (v[n-1]*v[n-3] - v[n-2]^2)/v[n-4]); concat([0], v) \\ G. C. Greubel, Sep 22 2018

PARI

{a(n) = my(E = ellinit([1, 0, 0, -2, 1]), z =  ellpointtoz(E, [1, -1])); round( ellsigma(E, n*z) / ellsigma(E, z)^n^2)}; /* Michael Somos, Sep 27 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. - Michael Somos, Aug 06 2014

0 = a(n)*a(n+5) - a(n+1)*a(n+4) + 2*a(n+2)*a(n+3) for all n in Z. - Michael Somos, Sep 27 2018

Extensions

Missing a(0)=0 and a(1)=1 added by Michael Somos, Aug 06 2014

More terms from Vincenzo Librandi, Aug 07 2014