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A208211 a(n)=(a(n-1)^2*a(n-2)^4+1)/a(n-3) with a(0)=a(1)=a(2)=1.

%I A208211 #20 Aug 26 2016 07:58:00
%S A208211 1,1,1,2,5,401,50250313,13058251494934169005517674,
%T A208211 2711319949800838662068317571116321157238013748056632969662193456875554487084437
%N A208211 a(n)=(a(n-1)^2*a(n-2)^4+1)/a(n-3) with a(0)=a(1)=a(2)=1.
%C A208211 This is the case a=4, 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 A208211 The next term, a(9), has 250 digits. - _Harvey P. Dale_, May 12 2015
%H A208211 Seiichi Manyama, <a href="/A208211/b208211.txt">Table of n, a(n) for n = 0..10</a>
%H A208211 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 A208211 From _Vaclav Kotesovec_, May 20 2015: (Start)
%F A208211 a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
%F A208211 d1 = -1.391382380630900845100729034616031832171938259539254240563846155543...
%F A208211 d2 = 0.2271344421706896320468868758105588761186297860618178147525916240716...
%F A208211 d3 = 3.1642479384602112130538421588054729560533084734774364258112545314714...
%F A208211 are the roots of the equation d^3 + 1 = 2*d^2 + 4*d and
%F A208211 c1 = 0.9492747639156309053009206968548726546571223067568220073025225799006...
%F A208211 c2 = 0.2025736158012536053359109009272747757676200151893348144191432397054...
%F A208211 c3 = 1.0182066570849459786725527422494583474915007718333213073686225606760...
%F A208211 (End)
%p A208211 a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)^4+1)/a(n-3): end: seq(a(i),i=0..10);
%t A208211 RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==(a[n-1]^2*a[n-2]^4+1)/a[n-3]},a,{n,9}] (* _Harvey P. Dale_, May 12 2015 *)
%Y A208211 Cf. A005246, A208204, A208210, A208212.
%K A208211 nonn
%O A208211 0,4
%A A208211 _Matthew C. Russell_, Apr 23 2012