A208203 a(n) = (a(n-1)*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 3, 25, 338, 1760417, 2719102918193, 43888992061611808973481301345, 501206842313618355048837897498360450999462416742984495192498

Offset: 0

Author

Matthew C. Russell, Apr 23 2012

Keywords

• nonn

Comments

This is the case a=3, b=1, y(0)=y(1)=y(2)=1 of the recurrence shown in the Example 3.2 of "The Laurent phenomenon" (see Link lines, p. 10).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..13
• Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001).
• Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, Advances in Applied Mathematics 28 (2002), 119-144.

Crossrefs

Cf. A005246, A208202, A208204, A208210.

Programs

Maple

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)*a(n-2)^3+1)/a(n-3): end: seq(a(i),i=0..10);

Mathematica

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (a[n - 1]*a[n - 2]^3 + 1)/a[n - 3];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 14 2017 *)
nxt[{a_,b_,c_}]:={b,c,(c*b^3+1)/a}; NestList[nxt,{1,1,1},10][[;;,1]] (* Harvey P. Dale, Nov 19 2023 *)

Formula

From Vaclav Kotesovec, May 20 2015: (Start)

a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where

d1 = -1.481194304092015622633537241216857180552745216998476728395893140813...

d2 = 0.3111078174659818999302814767914862551326055871751667747271657344269...

d3 = 2.1700864866260337227032557644253709254201396298233099536687274063868...

are the roots of the equation d^3 + 1 = d^2 + 3*d and

c1 = 0.9558632550121524723294926402589664329208850973886195977958538648966...

c2 = 0.0925177857987965285678801091508493414479538300221910521000975614673...

c3 = 1.0621981744880569938247885786471114069804924018378928906529142898259...

(End)