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

0,5

References

James Propp, personal communication.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..18

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A005246, A072713.

Sequence in context: A091771 A339806 A072714 * A133400 A113845 A072713

Adjacent sequences: A051783 A051784 A051785 * A051787 A051788 A051789

Keyword

nonn,nice,easy

Author

Michael Somos, Dec 09 1999

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007

Status

approved