A208202 -id:A208202 - cp's OEIS frontend

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

Matthew C. Russell, Apr 23 2012

nonn

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).

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.

Cf. A005246, A208202, A208206, A208210, A208213.

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);

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 *)

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)

A208203 a(n) = (a(n-1)*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 3, 25, 338, 1760417, 2719102918193, 43888992061611808973481301345, 501206842313618355048837897498360450999462416742984495192498

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=3, b=1, 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..13
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, A208202, A208204, A208210.

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)*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]*a[n - 2]^3 + 1)/a[n - 3];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 14 2017 *)
nxt[{a_,b_,c_}]:={b,c,(c*b^3+1)/a}; NestList[nxt,{1,1,1},10][[;;,1]] (* Harvey P. Dale, Nov 19 2023 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1.481194304092015622633537241216857180552745216998476728395893140813...
d2 = 0.3111078174659818999302814767914862551326055871751667747271657344269...
d3 = 2.1700864866260337227032557644253709254201396298233099536687274063868...
are the roots of the equation d^3 + 1 = d^2 + 3*d and
c1 = 0.9558632550121524723294926402589664329208850973886195977958538648966...
c2 = 0.0925177857987965285678801091508493414479538300221910521000975614673...
c3 = 1.0621981744880569938247885786471114069804924018378928906529142898259...
(End)

A208206 a(n)=(a(n-1)^2*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 5, 51, 6503, 431347892, 23724602128927104843, 37334625705205335653803036700733450756576803

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=1, 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(10)) has 98 digits. - Harvey P. Dale, Oct 04 2014

Seiichi Manyama, Table of n, a(n) for n = 0..12
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A005246, A208202, A208207, A208209.

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-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]^2 a[n-2]+1)/a[n-3]},a,{n,10}] (* Harvey P. Dale, Oct 04 2014 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -0.80193773580483825247220463901489010233183832426371430010712484639886484...
d2 = 0.554958132087371191422194871006410481067288862470910089376025968205157535...
d3 = 2.246979603717467061050009768008479621264549461792804210731098878193707304...
are the roots of the equation d^3 + 1 = 2*d^2 + d and
c1 = 0.874335057499939749225491691816700793966151250175012051621456437468590379...
c2 = 0.402356411273897640287204171338236092104516307383060911032953286637247174...
c3 = 1.071117422488325114038954501945557033632156032599675833309484054582086570...
(End)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A208203 a(n) = (a(n-1)*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A208206 a(n)=(a(n-1)^2*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula