A208213 a(n)=(a(n-1)^3*a(n-2)^2+1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 9, 2917, 1005227383127, 960336157066554685167071011848947459782832

Offset: 0

Author

Matthew C. Russell, Apr 23 2012

Keywords

• nonn

Comments

This is the case a=2, b=3, 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..9
• Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Crossrefs

Cf. A005246, A208207, A208209, A208214.

Programs

Maple

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

Mathematica

RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==(a[n-1]^3 a[n-2]^2+1)/a[n-3]},a,{n,10}] (* Harvey P. Dale, Jan 24 2014 *)

Formula

From Vaclav Kotesovec, May 20 2015: (Start)

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

d1 = -0.834243184313921717115626125802356204078143759301838339196857934562...

d2 = 0.3433795689528896338577674315423659679880371604828202900379886914176...

d3 = 3.4908636153610320832578586942599902360901065988190180491588692431448...

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

c1 = 0.8780803541847027315058502579763355822688533316057717751329965683549...

c2 = 0.4420233041946828357635108827822581168188691631054586381824944218534...

c3 = 1.0154140443448836210836588567949793209798883476847171784955774310427...

(End)