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%I #26 Apr 14 2022 13:37:00
%S 1,1,1,2,5,201,2525063,10355298070412763074,
%T 8589063344901709900442551790362661608528200120823830773
%N a(n)=(a(n-1)^2*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.
%C This is the case a=3, 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 The next term -- a(9) -- has 161 digits. - _Harvey P. Dale_, Apr 14 2022
%H Seiichi Manyama, <a href="/A208210/b208210.txt">Table of n, a(n) for n = 0..10</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 = -1.198691243515997113071999692569776193916276872472594369204332359716...
%F d2 = 0.2864620650316004980582127604312427653427138786836169481458128553091...
%F d3 = 2.9122291784843966150137869321385334285735629937889774210585195044073...
%F are the roots of the equation d^3 + 1 = 2*d^2 + 3*d and
%F c1 = 0.9326266928252752296152676800592959458631493222642463226349218269187...
%F c2 = 0.2535475214701961189033928082745089316567819534655391761010907360554...
%F c3 = 1.0248087086665041891835364490857429725941144848712661648932932629036...
%F (End)
%p a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)^3+1)/a(n-3): end: seq(a(i),i=0..10);
%t a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (a[n-1]^2*a[n-2]^3 + 1)/a[n-3];
%t Array[a, 10, 0] (* _Jean-François Alcover_, Dec 14 2017 *)
%t nxt[{a_,b_,c_}]:={b,c,(c^2 b^3+1)/a}; NestList[nxt,{1,1,1},10][[All,1]] (* _Harvey P. Dale_, Apr 14 2022 *)
%Y Cf. A005246, A208203, A208209, A208211, A208214.
%K nonn
%O 0,4
%A _Matthew C. Russell_, Apr 23 2012
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