A208210 a(n)=(a(n-1)^2*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 5, 201, 2525063, 10355298070412763074, 8589063344901709900442551790362661608528200120823830773

Offset

0,4

Comments

This is the case a=3, 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).

The next term -- a(9) -- has 161 digits. - Harvey P. Dale, Apr 14 2022

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10

Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001).

Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, Advances in Applied Mathematics 28 (2002), 119-144.

Formula

From Vaclav Kotesovec, May 20 2015: (Start)

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

d1 = -1.198691243515997113071999692569776193916276872472594369204332359716...

d2 = 0.2864620650316004980582127604312427653427138786836169481458128553091...

d3 = 2.9122291784843966150137869321385334285735629937889774210585195044073...

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

c1 = 0.9326266928252752296152676800592959458631493222642463226349218269187...

c2 = 0.2535475214701961189033928082745089316567819534655391761010907360554...

c3 = 1.0248087086665041891835364490857429725941144848712661648932932629036...

(End)

Maple

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)^3+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]^3 + 1)/a[n-3];
Array[a, 10, 0] (* Jean-François Alcover, Dec 14 2017 *)
nxt[{a_, b_, c_}]:={b, c, (c^2 b^3+1)/a}; NestList[nxt, {1, 1, 1}, 10][[All, 1]] (* Harvey P. Dale, Apr 14 2022 *)

Crossrefs

Cf. A005246, A208203, A208209, A208211, A208214.

Sequence in context: A012954 A006271 A013105 * A216458 A027682 A096695

Adjacent sequences: A208207 A208208 A208209 * A208211 A208212 A208213

Keyword

nonn

Author

Matthew C. Russell, Apr 23 2012

Status

approved