A208214 a(n)=(a(n-1)^3*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 9, 5833, 72339160083737, 8347449602301100278574002746114271427525770715131218

Offset: 0

Author

Matthew C. Russell, Apr 23 2012

Keywords

• nonn

Comments

This is the case a=3, 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, A208210, A208213.

Programs

Maple

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

Mathematica

RecurrenceTable[{a[n] == (a[n - 1]^3*a[n - 2]^3 + 1)/a[n - 3], a[0] == a[1] == a[2] == 1}, a, {n, 0, 7}] (* Michael De Vlieger, Mar 19 2017 *)

Formula

From Vaclav Kotesovec, May 20 2015: (Start)

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

d1 = -1

d2 = 2-sqrt(3) = 0.2679491924311227064725536584941276330571947461896193719...

d3 = 2+sqrt(3) = 3.7320508075688772935274463415058723669428052538103806280...

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

c1 = 0.9085343342123995498629194372995408229585378171837724081842452659181...

c2 = 0.3811823487030541690662698257664022175009714305688428757048879374472...

c3 = 1.0119167333492916399265234093841995850496968884402785055210058839859...

(End)