A208209 a(n) = (a(n-1)^2*a(n-2)^2 + 1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 5, 101, 127513, 33172764857794, 177153971843949087009428690473769185

Offset: 0

Author

Matthew C. Russell, Apr 23 2012

Keywords

• nonn

Comments

This is the case a=2, b=2, 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..11
• Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.
• 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, A208202, A208206, A208210, A208213.

Programs

Maple

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)^2+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]^2*a[n-2]^2 + 1)/a[n-3];
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Nov 17 2017 *)
nxt[{a_,b_,c_}]:={b,c,(c^2 b^2+1)/a}; NestList[nxt,{1,1,1},8][[;;,1]] (* Harvey P. Dale, Mar 24 2025 *)

Formula

From Vaclav Kotesovec, May 20 2015: (Start)

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

d1 = -1

d2 = (3-sqrt(5))/2 = 0.381966011250105151795413165634361882279690820194237...

d3 = (3+sqrt(5))/2 = 2.618033988749894848204586834365638117720309179805762...

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

c1 = 0.9084730936822995591913406002175634029260903950386034752117808169903...

c2 = 0.3198114201427769362008537317523839726550617444688426214134486371587...

c3 = 1.0375048945851318188473394167711806349224412339663566324740449820203...

(End)