Height growth of solutions and a discrete Painlevé equation

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Consider the discrete equation yₙ₊₁+yₙ₋₁=aₙ+bₙyₙ+cₙyₙ²/1-yₙ², where the right side is of degree two in yₙ and where the coefficients aₙ, bₙ and cₙ are rational functions of n with rational coefficients. Suppose that there is a solution such that for all sufficiently large n, yₙ and the height of yₙ dominates the height of the coefficient functions aₙ, bₙ and cₙ. We show that if the logarithmic height of yₙ 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 yₙ is also an admissible solution of a discrete Riccati equation. This provides further evidence that slow height growth is a good detector of integrability.