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

%I A208203 #27 Nov 19 2023 13:19:11
%S A208203 1,1,1,2,3,25,338,1760417,2719102918193,43888992061611808973481301345,
%T A208203 501206842313618355048837897498360450999462416742984495192498
%N A208203 a(n) = (a(n-1)*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.
%C A208203 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).
%H A208203 Seiichi Manyama, <a href="/A208203/b208203.txt">Table of n, a(n) for n = 0..13</a>
%H A208203 Sergey Fomin and Andrei Zelevinsky, <a href="https://arxiv.org/abs/math/0104241">The Laurent phenomenon</a>, arXiv:math/0104241v1 [math.CO] (2001).
%H A208203 Sergey Fomin and Andrei Zelevinsky, <a href="https://doi.org/10.1006/aama.2001.0770">The Laurent phenomenon</a>, Advances in Applied Mathematics 28 (2002), 119-144.
%F A208203 From _Vaclav Kotesovec_, May 20 2015: (Start)
%F A208203 a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
%F A208203 d1 = -1.481194304092015622633537241216857180552745216998476728395893140813...
%F A208203 d2 = 0.3111078174659818999302814767914862551326055871751667747271657344269...
%F A208203 d3 = 2.1700864866260337227032557644253709254201396298233099536687274063868...
%F A208203 are the roots of the equation d^3 + 1 = d^2 + 3*d and
%F A208203 c1 = 0.9558632550121524723294926402589664329208850973886195977958538648966...
%F A208203 c2 = 0.0925177857987965285678801091508493414479538300221910521000975614673...
%F A208203 c3 = 1.0621981744880569938247885786471114069804924018378928906529142898259...
%F A208203 (End)
%p A208203 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);
%t A208203 a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (a[n - 1]*a[n - 2]^3 + 1)/a[n - 3];
%t A208203 Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Dec 14 2017 *)
%t A208203 nxt[{a_,b_,c_}]:={b,c,(c*b^3+1)/a}; NestList[nxt,{1,1,1},10][[;;,1]] (* _Harvey P. Dale_, Nov 19 2023 *)
%Y A208203 Cf. A005246, A208202, A208204, A208210.
%K A208203 nonn
%O A208203 0,4
%A A208203 _Matthew C. Russell_, Apr 23 2012