On an observation of D'Ocagne concerning the fundamental sequence (Q2707844)

Let be \(\{W_n\}= \{W_n (a,b;p,q)\}\) the sequence defined by \(W_n= pW_{n-1} qW_{n-2}\), \(n\geq 2\), \(W_0=a\), \(W_1=b\). A sequence \(\{S_n\}= \{W_n (S_0,S_1; p,q)\}\) is said to have the property of D'Ocagne if there exist integers \(c_0\), \(c_1\) such that \(W_n= c_0 S_n+ c_1 S_{n+1}\), \(n\in \mathbb{Z}\). In this paper for \(q= \pm 1\) all sequences having the property of D'Ocagne are characterized.