Height growth of solutions and a discrete Painlevé equation

Consider the discrete equation y _n+1 + y _n-1 = a_n + b_ny_n + c_ny²_n/1-y²_n where the right side is of degree two in y_n and where the coefficients a_n, b_n and c_n are rational functions of n with rational coefficients. Suppose that there is a solution such that for all sufficiently large n, and the height of y_n dominates the height of the coefficient functions a_n, b_n and c_n. We show that if the logarithmic height of y_n grows no faster than a power of n then either the equation is a well known discrete Painlevé equation dP_II or its autonomous version or y_n is also an admissible solution of a discrete Riccati equation. This provides further evidence that slow height growth is a good detector of integrability.