A272073 - OEIS

A272073

Exponents of x in the numerator of cluster variables of rank 2 cluster algebras.

3, 6, 15, 24, 57, 90, 213, 336, 795, 1254, 2967, 4680, 11073, 17466, 41325, 65184, 154227, 243270, 575583, 907896, 2148105, 3388314, 8016837, 12645360, 29919243, 47193126, 111660135, 176127144, 416721297, 657315450, 1555225053, 2453134656, 5804178915

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OFFSET

2,1

COMMENTS

These numbers are the exponents of x in the numerator 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)=3, a(3)=6, a(4)=15, a(5)=24.

G.f.: 3*x^2*(1+x)^2 / (1-4*x^2+x^4). - Colin Barker, Apr 22 2016

PROG

(PARI) Vec(3*x^2*(1+x)^2/(1-4*x^2+x^4) + O(x^50)) \\ Colin Barker, Apr 22 2016