A178075 A (1,2) Somos-4 sequence.

0, 1, 1, -2, 1, 9, -16, -73, -145, 1442, 3951, -49121, 273184, 4095073, -13134671, -609740514, -7877081935, 206841213449, 153991537616, -138343392420553, 3626684171795871, 187758843981508546, 2146096698964614559

Offset: 0

Author

Paul Barry, May 19 2010

Keywords

• easy
• sign

Comments

Hankel transform of A178074 is a(n+2).

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -2, z = 1. - Michael Somos, Aug 06 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 0..150
• Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
• C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

Programs

Magma

I:=[0,1,1,-2,1]; [n le 5 select I[n] else (Self(n-1)*Self(n-3)+2*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] + 2*a[n - 2]^2)/a[n - 4], a[1] == 1, a[2] == 1, a[3] == -2, a[4] == 1}, a, {n, 1, 50}]] (* G. C. Greubel, Sep 22 2018 *)

PARI

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

Formula

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

a(-n) = -a(n). - Michael Somos, Aug 06 2014

Extensions

Added missing a(0)=0 and a(1)=1. - Michael Somos, Aug 06 2014