A208204 - OEIS

%I #28 Dec 14 2017 09:18:57

%S 1,1,1,2,3,49,1985,3814376662,1208563686390770296199,

%T 128885284912846137074628029815898112630258374651779168689

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

%C This is the case a=4, 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 Seiichi Manyama, <a href="/A208204/b208204.txt">Table of n, a(n) for n = 0..12</a>

%H Sergey Fomin and Andrei Zelevinsky, <a href="https://arxiv.org/abs/math/0104241">The Laurent phenomenon</a>, arXiv:math/0104241v1 [math.CO] (2001).

%H Sergey Fomin and Andrei Zelevinsky, <a href="https://doi.org/10.1006/aama.2001.0770">The Laurent phenomenon</a>, 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 = -1.699628148275317956229728291667145232598924547592878096541472700997...

%F d2 = 0.2391232782565544642500835033134825869161430421361867747730632704531...

%F d3 = 2.4605048700187634919796447883536626456827815054566913217684094305444...

%F are the roots of the equation d^3 + 1 = d^2 + 4*d and

%F c1 = 0.9668824482256124500532459849115781952211866063916062435395239896336...

%F c2 = 0.0680423294122660088493946488133224274885942757072304155092839505634...

%F c3 = 1.0386083844527725102069795872299989830277012965629707721463998933768...

%F (End)

%p a:=proc(n) if n<3 then return 1: fi: return (a(n-1)*a(n-2)^4+1)/a(n-3): end: seq(a(i),i=1..10);

%t a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (a[n - 1]*a[n - 2]^4 + 1)/a[n - 3];

%t Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Dec 14 2017 *)

%Y Cf. A005246, A208203, A208205, A208211.

%K nonn

%O 0,4

%A _Matthew C. Russell_, Apr 23 2012