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A208202
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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.
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4
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1, 1, 1, 2, 3, 13, 59, 3324, 890065, 166683166499, 39725939269090918399, 1240040687243304530118746458657221560, 11740660815927242416329935330676365456512664243108711550072429939
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OFFSET
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0,4
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COMMENTS
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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).
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LINKS
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Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.
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FORMULA
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a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1.24697960371746706105000976800847962126454946179280421073109887819...
d2 = 0.445041867912628808577805128993589518932711137529089910623974031794...
d3 = 1.801937735804838252472204639014890102331838324263714300107124846398...
are the roots of the equation d^3 + 1 = d^2 + 2*d and
c1 = 0.937508205283971584227188160392119895660526011507051773879367647962...
c2 = 0.127128212809518009874462927372545164747593272064601714573478901156...
c3 = 1.135040592200579625529345655593495454581148721169010026906480955795...
(End)
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MATHEMATICA
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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 *)
(* 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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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