A208207 -id:A208207 - cp's OEIS frontend

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

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)

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)

A208208 a(n)=(a(n-1)^4*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 17, 167043, 6618080569762280805809

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=1, b=4, 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..8
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.

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^4*a(n-2)+1)/a(n-3): end: seq(a(i),i=0..10);

RecurrenceTable[{a[n] == (a[n - 1]^4*a[n - 2] + 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 = -0.588363990685104156421284586508527584304318862407786509166141051262...
d2 = 0.4064206546327112651910488344937800073049991477253475806754539682375...
d3 = 4.1819433360523928912302357520147475769993197146824389284906870830246...
are the roots of the equation d^3 + 1 = 4*d^2 + d and
c1 = 0.8094826741348488413005600397911253102639462301397489110738060562305...
c2 = 0.5758908197062035276668941188013698534573120455706764136847247903030...
c3 = 1.0094396347013780675988108222508397688561313671701492219003321772184...
(End)

A258161 a(n) = a(n-1)^3/a(n-2) with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 8, 256, 2097152, 36028797018963968, 22300745198530623141535718272648361505980416

Offset: 0

Vaclav Kotesovec, May 22 2015

The next term has 114 digits.

Cf. A000045, A000301, A001906, A064098, A208207, A230900, A258169.

Clear[a]; RecurrenceTable[{a[n]==a[n-1]^3/a[n-2], a[0]==1, a[1]==2},a,{n,0,8}]
Clear[a]; a[0]=2; a[n_]:=a[n]=Product[a[j]^(n-j),{j,0,n-1}]; Flatten[{1, Table[a[n], {n,1,8}]}]
Table[2^(Fibonacci[2*n]), {n, 0, 8}]

a(n) = 2^(Fibonacci(2*n)).

cp's OEIS Frontend

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

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

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A208208 a(n)=(a(n-1)^4*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

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A258161 a(n) = a(n-1)^3/a(n-2) with a(0)=1, a(1)=2.

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