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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.

%I A208206 #20 Mar 19 2017 08:46:39
%S A208206 1,1,1,2,5,51,6503,431347892,23724602128927104843,
%T A208206 37334625705205335653803036700733450756576803
%N A208206 a(n)=(a(n-1)^2*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.
%C A208206 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).
%C A208206 The next term (a(10)) has 98 digits. - _Harvey P. Dale_, Oct 04 2014
%H A208206 Seiichi Manyama, <a href="/A208206/b208206.txt">Table of n, a(n) for n = 0..12</a>
%H A208206 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 A208206 From _Vaclav Kotesovec_, May 20 2015: (Start)
%F A208206 a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
%F A208206 d1 = -0.80193773580483825247220463901489010233183832426371430010712484639886484...
%F A208206 d2 = 0.554958132087371191422194871006410481067288862470910089376025968205157535...
%F A208206 d3 = 2.246979603717467061050009768008479621264549461792804210731098878193707304...
%F A208206 are the roots of the equation d^3 + 1 = 2*d^2 + d and
%F A208206 c1 = 0.874335057499939749225491691816700793966151250175012051621456437468590379...
%F A208206 c2 = 0.402356411273897640287204171338236092104516307383060911032953286637247174...
%F A208206 c3 = 1.071117422488325114038954501945557033632156032599675833309484054582086570...
%F A208206 (End)
%p A208206 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);
%t A208206 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 *)
%Y A208206 Cf. A005246, A208202, A208207, A208209.
%K A208206 nonn
%O A208206 0,4
%A A208206 _Matthew C. Russell_, Apr 23 2012