A157101 A Somos-4 variant.

1, -1, -5, -4, 29, 129, -65, -3689, -16264, 113689, 2382785, 7001471, -398035821, -7911171596, 43244638645, 6480598259201, 124106986093951, -5987117709349201, -541051130050800400, -4830209396684261199

Offset: 0

Author

Paul Barry, Feb 22 2009

Keywords

• easy
• sign

Comments

Hankel transform of A157100.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..145
• Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Crossrefs

Cf. A051138.

Cf. A157005, A162546, A162547.

Programs

GAP

a:=[1,-1,-5,-4];; for n in [5..20] do a[n]:=(a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019

Magma

I:=[1,-1,-5,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019

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]==-5, a[3]==-4}, a, {n,20}] (* G. C. Greubel, Feb 23 2019 *)

PARI

m=20; v=concat([1,-1,-5,-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, Feb 23 2019

Sage

def a(n):
    if (n==0): return 1
    elif (n==1): return -1
    elif (n==2): return -5
    elif (n==3): return -4
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in (0..20)] # G. C. Greubel, Feb 23 2019

Formula

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), with a(0)=1, a(1)=-1, a(2)=-5, a(3)=-4.

a(n) = A051138(n+1) for all n in Z. - Michael Somos, Jul 17 2016