login
A208209
a(n) = (a(n-1)^2*a(n-2)^2 + 1)/a(n-3) with a(0)=a(1)=a(2)=1.
7
1, 1, 1, 2, 5, 101, 127513, 33172764857794, 177153971843949087009428690473769185
OFFSET
0,4
COMMENTS
This is the case a=2, 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).
LINKS
Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001); Advances in Applied Mathematics 28 (2002), 119-144.
FORMULA
From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1
d2 = (3-sqrt(5))/2 = 0.381966011250105151795413165634361882279690820194237...
d3 = (3+sqrt(5))/2 = 2.618033988749894848204586834365638117720309179805762...
are the roots of the equation d^3 + 1 = 2*d^2 + 2*d and
c1 = 0.9084730936822995591913406002175634029260903950386034752117808169903...
c2 = 0.3198114201427769362008537317523839726550617444688426214134486371587...
c3 = 1.0375048945851318188473394167711806349224412339663566324740449820203...
(End)
MAPLE
a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)^2+1)/a(n-3): end: seq(a(i), i=0..10);
MATHEMATICA
a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (a[n-1]^2*a[n-2]^2 + 1)/a[n-3];
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Nov 17 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthew C. Russell, Apr 23 2012
STATUS
approved