A208213 -id:A208213

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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.

+10
7

1, 1, 1, 2, 5, 101, 127513, 33172764857794, 177153971843949087009428690473769185

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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).

LINKS

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.

FORMULA

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)

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Matthew C. Russell, Apr 23 2012

STATUS

approved

A208207

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

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.

FORMULA

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)

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

Cf. A005246, A208206, A208208, A208213.

KEYWORD

nonn

AUTHOR

Matthew C. Russell, Apr 23 2012

STATUS

approved

A208214

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

+10
3

1, 1, 1, 2, 9, 5833, 72339160083737, 8347449602301100278574002746114271427525770715131218

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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).

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.

FORMULA

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)

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

Cf. A005246, A208210, A208213.

KEYWORD

nonn

AUTHOR

Matthew C. Russell, Apr 23 2012

STATUS

approved

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