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A208212 a(n) = (a(n-1)^2*a(n-2)^5+1)/a(n-3) with a(0)=a(1)=a(2)=1.

%I A208212 #23 Nov 25 2017 15:45:24
%S A208212 1,1,1,2,5,801,1002501563,66276977238296815913344374183794
%N A208212 a(n) = (a(n-1)^2*a(n-2)^5+1)/a(n-3) with a(0)=a(1)=a(2)=1.
%C A208212 This is the case a=5, 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 A208212 Seiichi Manyama, <a href="/A208212/b208212.txt">Table of n, a(n) for n = 0..9</a>
%H A208212 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 A208212 From _Vaclav Kotesovec_, May 20 2015: (Start)
%F A208212 a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
%F A208212 d1 = -1.575773472651936072015953246349296378313356749177416595434978648425...
%F A208212 d2 = 0.1872837251102239188569922313039458439968721185362219238420888422761...
%F A208212 d3 = 3.3884897475417121531589610150453505343164846306411946715928898061494...
%F A208212 are the roots of the equation d^3 + 1 = 2*d^2 + 5*d and
%F A208212 c1 = 0.9607631794694254165284953988161129828633931861764073755339129251426...
%F A208212 c2 = 0.1625672201779811599302887070429221376610589038410298300467412998556...
%F A208212 c3 = 1.0141969317515019907302101637404918873873074910913934972790303073225...
%F A208212 (End)
%p A208212 a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)^5+1)/a(n-3): end: seq(a(i),i=0..10);
%t A208212 a[n_] := a[n] = If[n <= 2, 1, (a[n - 1]^2*a[n - 2]^5 + 1)/a[n - 3]];
%t A208212 Table[a[n], {n, 0, 7}] (* _Jean-François Alcover_, Nov 25 2017 *)
%Y A208212 Cf. A005246, A208211.
%K A208212 nonn
%O A208212 0,4
%A A208212 _Matthew C. Russell_, Apr 23 2012