  D. S. Hong and P. Pongsriiam have provided a necessary and sufficient
condition for the generating function for Fibonacci numbers (resp. the Lucas
numbers) to be an integer value, for rational numbers. In other words, their
results relate to the integer values of the generating functions of the
sequences obtained from the integer solutions of Pell's equation
$5x^{2}-y^{2}=\pm4$. If we change this Pell's equation to another type of
Pell's equation, how will their results change? This is a natural and
interesting problem. In this paper, we show that a result similar to theirs is
obtained for the generating functions for sequences given by Pell's equation
$x^{2}-my^{2}=\pm1 \ (m\text{ is a non-square natural number})$.
