The Integer Sequence \(B = B_n (P, Q)\) with Parameters \(P\) and \(Q\)

Abstract

In this work, we first prove that every prime number \(p \equiv 1 \pmod{4}\) can be written in the form \(p = P^2 – 4Q\)with two positive integers \(P\) and \(Q\). Then, we define the sequence \(B_n(P, Q)\) to be \(B_0 = 2\), \(B_1 = P\), and \(B_n = PB_{n-1} – QB_{n-2}\) for \(n \geq 2\), and derive some algebraic identities on it. Also, we formulate the limit of the cross-ratio for four consecutive numbers \(B_n\), \(B_{n+1}\), \(B_{n+2}\), and \(B_{n+3}\).