# Prove that a holonomic (p-recursive) difference equation returns only integral values

Consider the recurrence given by

$(n+1)^2 a_{n+1} = (9n^2+9n+3)a_n-27n^2 a_{n-1}$

$a_0 = 1, a_1 = 3$.

Clearly, $a_n$ is rational, but unexpectedly, the recurrence seems to output only integral values. Can you prove that this is indeed so? For more terms, see A006077 in the OEIS.

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Did you follow up the reference given at the OEIS? –  Gerry Myerson Jun 14 '11 at 1:13

This looks a lot like Apery's recurrence $$(n+1)^2a_{n+1}=(11n^2+11n+3)a_n+n^2a_{n-1}$$ Many papers have been written on generalizations of Apery's recurrence. I don't know whether any of them have studied the particular recurrence of this question, but it might be worth having a look at