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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).

%I A051786 #46 Feb 19 2024 01:58:30
%S A051786 1,1,1,1,2,3,7,43,452,45351,125920291,60027819184831,
%T A051786 758397193749171922281611,126403219004744354228963383975713263866432,
%U A051786 45699526286117471520994956894648733172150425791690122432447239675853643
%N 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).
%D A051786 _James Propp_, personal communication.
%H A051786 Seiichi Manyama, <a href="/A051786/b051786.txt">Table of n, a(n) for n = 0..18</a>
%F A051786 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).
%F A051786 a(-n) = a(n+3).
%F A051786 From _Vaclav Kotesovec_, May 20 2015: (Start)
%F A051786 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
%F A051786 c1 = 0.0858378165313271469223136812741638183980800626360336156811045938771...
%F A051786 c2 = 1.0479981158737678235689040669973933524451313410375783562899638042343...
%F A051786 c3 = 1.0681060454695696105471945019699938961207077685059613621050203396954...
%F A051786 c4 = 0.0530316436302789163635657674741144158928386126460043035284221194603...
%F A051786 (End)
%t A051786 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 *)
%o A051786 (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 */
%o A051786 (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
%o A051786 (Haskell)
%o A051786 a051786 n = a051786_list !! n
%o A051786 a051786_list = 1 : 1 : 1 : 1 :
%o A051786    zipWith div (tail $ zipWith3 (\u v w -> 1 + u * v * w)
%o A051786                (drop 2 a051786_list) (tail a051786_list) a051786_list)
%o A051786                a051786_list
%o A051786 -- _Reinhard Zumkeller_, Jan 07 2014
%Y A051786 Cf. A005246, A072713.
%K A051786 nonn,nice,easy
%O A051786 0,5
%A A051786 _Michael Somos_, Dec 09 1999
%E A051786 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 17 2007