OFFSET
2,2
COMMENTS
These numbers are the highest exponents of x in the numerators of cluster variables x(n) for n > 1 of the cluster algebra A(b,c) of rank 2 with parameters b=2 and c=3. Recall that the algebra A(2,3) is a subalgebra of Q(x,y) generated by the following infinite family of variables:
x(n) = (1+x(n-1)^2)/x(n-2) if n is even,
x(n) = (1+x(n-1)^3)/x(n-2) if n is odd,
with x(0)=x and x(1)=y.
LINKS
Colin Barker, Table of n, a(n) for n = 2..1000
Andrei Zelevinsky, What is a Cluster Algebra?, AMS Notices 54(11): 1494-1495, (2007).
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).
FORMULA
a(n) = 4*a(n-2)-a(n-4), with a(2)=0, a(3)=3, a(4)=6, a(5)=15.
G.f.: 3*x^3*(1+x)^2 / (1-4*x^2+x^4). - Colin Barker, Apr 22 2016
EXAMPLE
x(2) = (1+y^2)/x, thus a(2) = 0.
x(3) = (x^3+1+3y^2+3y^4+y^6)/(x^3*y), thus a(3) = 3.
MATHEMATICA
LinearRecurrence[{0, 4, 0, -1}, {0, 3, 6, 15}, 50] (* Paolo Xausa, Sep 29 2025 *)
PROG
(PARI) Vec(3*x^3*(1+x)^2/(1-4*x^2+x^4) + O(x^50)) \\ Colin Barker, Apr 22 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Hector J. Blandin N., Apr 19 2016
EXTENSIONS
a(2) inserted by Andrei Zabolotskii, Sep 28 2025
STATUS
approved