A208208 -id:A208208 - cp's OEIS frontend

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

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)

A258169 a(n) = a(n-1)^4/a(n-2) with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 16, 32768, 72057594037927936, 822752278660603021077484591278675252491367932816789931674304512

Offset: 0

Vaclav Kotesovec, May 22 2015

The next term has 235 digits.

In general, the recurrence a(n) = a(n-1)^k / a(n-2) with a(0) = 1, a(1) = m, k > 2, has a solution a(n) = m^(((k+sqrt(k^2-4))^n - (k-sqrt(k^2-4))^n) / (sqrt(k^2-4) * 2^n)).

Cf. A001353, A112969, A208208, A258161.

RecurrenceTable[{a[n]==a[n-1]^4/a[n-2], a[0]==1, a[1]==2},a,{n,0,6}]
nxt[{a_,b_}]:={b,b^4/a}; NestList[nxt,{1,2},5][[All,1]] (* Harvey P. Dale, Sep 04 2022 *)

a(n) = 2^(A001353(n)).

a(n) = 2^(((2+sqrt(3))^n-(2-sqrt(3))^n)/(2*sqrt(3))).

cp's OEIS Frontend

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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