A208212 a(n) = (a(n-1)^2*a(n-2)^5+1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 5, 801, 1002501563, 66276977238296815913344374183794

Offset: 0

Author

Matthew C. Russell, Apr 23 2012

Keywords

• nonn

Comments

This is the case a=5, 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..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, A208211.

Programs

Maple

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)^5+1)/a(n-3): end: seq(a(i),i=0..10);

Mathematica

a[n_] := a[n] = If[n <= 2, 1, (a[n - 1]^2*a[n - 2]^5 + 1)/a[n - 3]];
Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Nov 25 2017 *)

Formula

From Vaclav Kotesovec, May 20 2015: (Start)

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

d1 = -1.575773472651936072015953246349296378313356749177416595434978648425...

d2 = 0.1872837251102239188569922313039458439968721185362219238420888422761...

d3 = 3.3884897475417121531589610150453505343164846306411946715928898061494...

are the roots of the equation d^3 + 1 = 2*d^2 + 5*d and

c1 = 0.9607631794694254165284953988161129828633931861764073755339129251426...

c2 = 0.1625672201779811599302887070429221376610589038410298300467412998556...

c3 = 1.0141969317515019907302101637404918873873074910913934972790303073225...

(End)