Suites récurrentes linéaires d'ordre 2 à divisibilité forte

Abstract

We reprove twice, in a simpler but as elementary way, a result by Horák and Skula (1985) who determined, among all sequences of integers defined by $$u_1=1,u_2=R,u_{n+2}=P{u}_{n+1}-Q{u}_n$$ for some integers $P,Q,R$, those which satisfy the strong divisibility condition $$\forall i,j\in {\mathbb{N}}^{\ast }{u}_i\wedge u_j=\left|u_{i\wedge j}\right|,$$ where $\wedge$ denotes the greatest common divisor.

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