A051786 Propp's cubic recurrence: a(0)=a(1)=a(2)=a(3)=1; for n>3, a(n)=(1+a(n-1)*a(n-2)*a(n-3))/a(n-4).
1, 1, 1, 1, 2, 3, 7, 43, 452, 45351, 125920291, 60027819184831, 758397193749171922281611, 126403219004744354228963383975713263866432, 45699526286117471520994956894648733172150425791690122432447239675853643
Offset: 0
References
- James Propp, personal communication.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..18
Programs
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Haskell
a051786 n = a051786_list !! n a051786_list = 1 : 1 : 1 : 1 : zipWith div (tail $ zipWith3 (\u v w -> 1 + u * v * w) (drop 2 a051786_list) (tail a051786_list) a051786_list) a051786_list -- Reinhard Zumkeller, Jan 07 2014 -
Mathematica
RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[n]==(1+a[n-1]a[n-2]a[n-3])/ a[n-4]},a[n],{n,15}] (* Harvey P. Dale, May 14 2011 *) -
PARI
{a(n) = if( n<0, n = 3-n); if( n<4, 1, (a(n-1) * a(n-2) * a(n-3) + 1) / a(n-4)) } /* Michael Somos, Oct 16 2006 */ -
PARI
a=vector(15); a[1]=a[2]=a[3]=1;a[4]=2; for(n=5, #a, a[n]=(1+a[n-1]*a[n-2]*a[n-3])/a[n-4]); concat(1, a) \\ Altug Alkan, Sep 27 2018
Formula
a(0)=a(1)=a(2)=a(3)=1; for n>3, a(n+2)*a(n-2) = 1 + a(n+1)*a(n)*a(n-1).
a(-n) = a(n+3).
From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(((1+sqrt(13)-sqrt(2*sqrt(13)-2))/4)^n) * c2^(((1+sqrt(13)+sqrt(2*sqrt(13)-2))/4)^n) * (c3^2+c4^2)^((-1)^n * cos(n*arccot(sqrt((2*sqrt(13)-5)/3)))) * exp(2*(-1)^n*arctan(c4/c3) * sin(n*arccot(sqrt((2*sqrt(13)-5)/3)))), where
c1 = 0.0858378165313271469223136812741638183980800626360336156811045938771...
c2 = 1.0479981158737678235689040669973933524451313410375783562899638042343...
c3 = 1.0681060454695696105471945019699938961207077685059613621050203396954...
c4 = 0.0530316436302789163635657674741144158928386126460043035284221194603...
(End)
Extensions
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007
Comments