A208213 -id:A208213 - 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)

A208207 a(n)=(a(n-1)^3*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 9, 1459, 13975855106, 442535332406378982945622818194705

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=1, 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).

The next term has 105 digits. - Harvey P. Dale, Jul 04 2022

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, A208206, A208208, A208213.

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

RecurrenceTable[{a[n] == (a[n - 1]^3*a[n - 2] + 1)/a[n - 3], a[0] == a[1] == a[2] == 1}, a, {n, 0, 8}] (* Michael De Vlieger, Mar 19 2017 *)
nxt[{a_, b_, c_}] := {b, c, (c^3 b + 1)/a}; NestList[nxt,{1,1,1},10][[All,1]] (* Harvey P. Dale, Jul 04 2022 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -0.675130870566646070889621798150060480808032527677372732612153869841...
d2 = 0.4608111271891108834741240973014799919001128904578732982807715533323...
d3 = 3.2143197433775351874154977008485804889079196372194994343313823165091...
are the roots of the equation d^3 + 1 = 3*d^2 + d and
c1 = 0.8399660110229591295951614867364338523629139731316529610703364786466...
c2 = 0.5166029105674572719002224224720428001985297645051505025129589573676...
c3 = 1.0214282112585594227681235564690028577352359049566082298453239674712...
(End)

A208214 a(n)=(a(n-1)^3*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, 9, 5833, 72339160083737, 8347449602301100278574002746114271427525770715131218

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=3, 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, A208210, A208213.

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

RecurrenceTable[{a[n] == (a[n - 1]^3*a[n - 2]^3 + 1)/a[n - 3], a[0] == a[1] == a[2] == 1}, a, {n, 0, 7}] (* Michael De Vlieger, Mar 19 2017 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1
d2 = 2-sqrt(3) = 0.2679491924311227064725536584941276330571947461896193719...
d3 = 2+sqrt(3) = 3.7320508075688772935274463415058723669428052538103806280...
are the roots of the equation d^3 + 1 = 3*d^2 + 3*d and
c1 = 0.9085343342123995498629194372995408229585378171837724081842452659181...
c2 = 0.3811823487030541690662698257664022175009714305688428757048879374472...
c3 = 1.0119167333492916399265234093841995850496968884402785055210058839859...
(End)

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

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A208207 a(n)=(a(n-1)^3*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

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A208214 a(n)=(a(n-1)^3*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

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Keywords

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Crossrefs

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Maple

Mathematica

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