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

%I #28 Sep 08 2022 08:46:01

%S 1,1,1,2,3,13,59,3324,890065,166683166499,39725939269090918399,

%T 1240040687243304530118746458657221560,

%U 11740660815927242416329935330676365456512664243108711550072429939

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

%C This is the case a=2, 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 Bruno Berselli, <a href="/A208202/b208202.txt">Table of n, a(n) for n = 0..16</a>

%H 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 From _Vaclav Kotesovec_, May 20 2015: (Start)

%F a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where

%F d1 = -1.24697960371746706105000976800847962126454946179280421073109887819...

%F d2 = 0.445041867912628808577805128993589518932711137529089910623974031794...

%F d3 = 1.801937735804838252472204639014890102331838324263714300107124846398...

%F are the roots of the equation d^3 + 1 = d^2 + 2*d and

%F c1 = 0.937508205283971584227188160392119895660526011507051773879367647962...

%F c2 = 0.127128212809518009874462927372545164747593272064601714573478901156...

%F c3 = 1.135040592200579625529345655593495454581148721169010026906480955795...

%F (End)

%t RecurrenceTable[{a[0] == a[1] == a[2] == 1, a[n] == (a[n - 1] a[n - 2]^2 + 1)/a[n - 3]}, a, {n, 12}] (* _Bruno Berselli_, Apr 25 2012 *)

%t (* The numerical values of the constants d1, d2, d3 *) Print[N[{Root[1-2*#1-#1^2+#1^3&,1], Root[1-2*#1-#1^2+#1^3&,2], Root[1-2*#1-#1^2+#1^3&,3]}, 80]]; (* and the constants c1, c2, c3 *) A208202 = RecurrenceTable[{a[0]==a[1]==a[2]==N[1,100], a[n] == (a[n-1]*a[n-2]^2 + 1)/a[n-3]}, a, {n,1,30}]; Table[Flatten[N[{Exp[cc1], Exp[cc2], Exp[cc3]}/.Solve[Table[Log[A208202[[n]]] == cc1*Root[1 - 2*#1 - #1^2 + #1^3&, 1]^n + cc2*Root[1 - 2*#1 - #1^2 + #1^3&, 2]^n + cc3*Root[1 - 2*#1 - #1^2 + #1^3&, 3]^n, {n, k, k+2}]],80]], {k, Length[A208202]-3, Length[A208202]-2}] (* _Vaclav Kotesovec_, May 20 2015 *)

%o (Magma) [n le 3 select 1 else (Self(n-1)*Self(n-2)^2+1)/Self(n-3): n in [1..13]]; // _Bruno Berselli_, Apr 24 2012

%Y Cf. A005246, A208203, A208209.

%K nonn

%O 0,4

%A _Matthew C. Russell_, Apr 23 2012