Let [tex]F_{n}[/tex] be the Fibonacci number, and let [tex]a_n=F_n^2[/tex]

Hadamard's product theorem implies that the generating function for the sequence [tex]{a_n}[/tex] must be rational.
By theorem 2, which states, "Assume the notations and conventions above. Then the sequence {a_n}[/tex] satisfies a linear recurrence relation
[tex]a_{n+k}=c_1a_{n+k-1}+c_2a_{n+k-2}+...+c_ka_n[/tex] for n ≥ 0"

The first few terms of the sequence {a_n}[/tex] are easy to calculate:
1,1,4,9,25,64,169,441,1156,3025,7921,20736,54289,….

Please check your relation with these numbers.