MathDB

Let

\{a_n \}_{n=0}^{\infty}

be a sequence given recrusively such that

a_0=1

and

a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2}

for

n\geq 0

Show that : a)

a_n

is a positive integer. b)

a_n a_{n+1}-1

is a square of an integer.
Proposed by Stefan Gyurki (Matej Bel University, Banska Bystrica).

Problem Statement