A208208 a(n)=(a(n-1)^4*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 17, 167043, 6618080569762280805809

Offset: 0

Author

Matthew C. Russell, Apr 23 2012

Keywords

• nonn

Comments

This is the case a=1, b=4, 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..8
• 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.

Programs

Maple

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^4*a(n-2)+1)/a(n-3): end: seq(a(i),i=0..10);

Mathematica

RecurrenceTable[{a[n] == (a[n - 1]^4*a[n - 2] + 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 = -0.588363990685104156421284586508527584304318862407786509166141051262...

d2 = 0.4064206546327112651910488344937800073049991477253475806754539682375...

d3 = 4.1819433360523928912302357520147475769993197146824389284906870830246...

are the roots of the equation d^3 + 1 = 4*d^2 + d and

c1 = 0.8094826741348488413005600397911253102639462301397489110738060562305...

c2 = 0.5758908197062035276668941188013698534573120455706764136847247903030...

c3 = 1.0094396347013780675988108222508397688561313671701492219003321772184...

(End)