A208214 -id:A208214 - cp's OEIS frontend

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

Matthew C. Russell, Apr 23 2012

nonn

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

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.

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

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);

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 *)

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)

A208213 a(n)=(a(n-1)^3*a(n-2)^2+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 9, 2917, 1005227383127, 960336157066554685167071011848947459782832

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=2, 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).

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.

Cf. A005246, A208207, A208209, A208214.

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

RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==(a[n-1]^3 a[n-2]^2+1)/a[n-3]},a,{n,10}] (* Harvey P. Dale, Jan 24 2014 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -0.834243184313921717115626125802356204078143759301838339196857934562...
d2 = 0.3433795689528896338577674315423659679880371604828202900379886914176...
d3 = 3.4908636153610320832578586942599902360901065988190180491588692431448...
are the roots of the equation d^3 + 1 = 3*d^2 + 2*d and
c1 = 0.8780803541847027315058502579763355822688533316057717751329965683549...
c2 = 0.4420233041946828357635108827822581168188691631054586381824944218534...
c3 = 1.0154140443448836210836588567949793209798883476847171784955774310427...
(End)

cp's OEIS Frontend

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

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