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The one-parameter family of second order nonlinear difference equations eachof which is given by $$ x_{n-1}x_nx_{n+1}=x_{n-1}+(x_n)^{\beta-1}+x_{n+1}\qquad(\beta\in\mathbb{N}) $$ is explored. Since the equation above is arisingfrom seed mutations of a rank 2 cluster algebra, its solution is periodic onlywhen $\beta\leq3$. In order to evaluate the dynamics with $\beta\geq4$,algebraic entropy of the birational map equivalent to the difference equationis investigated; it vanishes when $\beta=4$ but is positive when $\beta\geq5$.This fact suggests that the difference equation with $\beta\leq4$ is integrablebut that with $\beta\geq5$ is not. It is moreover shown that the differenceequation with $\beta\geq4$ fails the singularity confinement test. This fact isconsistent with linearizability of the equation with $\beta=4$ and reinforcesnon-integrability of the equation with $\beta\geq5$.

A family of integrable and non-integrable difference equations arising from cluster algebras