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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.

%I A208209 #39 Mar 24 2025 15:30:20
%S A208209 1,1,1,2,5,101,127513,33172764857794,
%T A208209 177153971843949087009428690473769185
%N A208209 a(n) = (a(n-1)^2*a(n-2)^2 + 1)/a(n-3) with a(0)=a(1)=a(2)=1.
%C A208209 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).
%H A208209 Seiichi Manyama, <a href="/A208209/b208209.txt">Table of n, a(n) for n = 0..11</a>
%H A208209 Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, <a href="https://doi.org/10.1007/s10801-015-0647-5">Laurent phenomenon sequences</a>, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.
%H A208209 Sergey Fomin and Andrei Zelevinsky, <a href="http://arxiv.org/abs/math/0104241">The Laurent phenomenon</a>, arXiv:math/0104241v1 [math.CO] (2001); Advances in Applied Mathematics 28 (2002), 119-144.
%F A208209 From _Vaclav Kotesovec_, May 20 2015: (Start)
%F A208209 a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
%F A208209 d1 = -1
%F A208209 d2 = (3-sqrt(5))/2 = 0.381966011250105151795413165634361882279690820194237...
%F A208209 d3 = (3+sqrt(5))/2 = 2.618033988749894848204586834365638117720309179805762...
%F A208209 are the roots of the equation d^3 + 1 = 2*d^2 + 2*d and
%F A208209 c1 = 0.9084730936822995591913406002175634029260903950386034752117808169903...
%F A208209 c2 = 0.3198114201427769362008537317523839726550617444688426214134486371587...
%F A208209 c3 = 1.0375048945851318188473394167711806349224412339663566324740449820203...
%F A208209 (End)
%p A208209 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);
%t A208209 a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (a[n-1]^2*a[n-2]^2 + 1)/a[n-3];
%t A208209 Table[a[n], {n, 0, 11}] (* _Jean-François Alcover_, Nov 17 2017 *)
%t A208209 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 *)
%Y A208209 Cf. A005246, A208202, A208206, A208210, A208213.
%K A208209 nonn
%O A208209 0,4
%A A208209 _Matthew C. Russell_, Apr 23 2012