Stability of two-term recurrence sequences with even parameter
Let \(a,b\) be fixed integers and let \((u_i)^\infty_{i=0}\) be the second-order linear recurring sequence \(u_i=au_{i-1} +bu_{i-2}\) with initial values \(u_0=0\), \(u_1=1\). For any positive integer \(m\), consider the corresponding sequence \((\overline u_i)\) of residue classes \(\text{mod } m\). If \(b\) and \(m\) are relatively prime, then \((\overline u_i)\) is purely periodic. For each integer \(d\), let \(\nu(m,d)\) denote the number of occurrences of the residue \(d\pmod m\) in one (shortest) period corresponding to a fixed recurring sequence \((u_i)\) and modulus \(m\). Set \[ \Omega(m)= \{\gamma (m,d): d\in \mathbb{Z}\}. \] The sequence is called stable modulo a prime \(p\) if there is a positive integer \(N\) such that \(\Omega (p^k)= \Omega (p^N)\) for all \(k\geq N\). In this paper the stability of sequences with even parameter \(a\) is characterized. The tools of the proofs are similar to those used by the authors in [\textit{W. Carlip} and \textit{E. Jacobson}, Finite Fields Appl. 2, 369-406 (1996)], where sequences \((u_i)\) with odd parameter \(a\) have been considered.
stability of sequences with even parameter
scientific article; zbMATH DE number 1199988
scientific article