Common values of linear recurrences related to Shank's simplest cubics

Abstract

Let $A,B,C\in Z$ not all zeroes and let $F(u,n)=F(A,B,C,u,n)$ be the linear recursive sequence, which is defined by the initial terms $F(u,0)=A,F(u,1)=B,F(u,2)=C$ and whose characteristic polynomial is Daniel Shanks simplest cubic $S_u\left(X\right)=X^3-(u-1)X^2-(u+2)X-1,u\in Z$. We prove that there exists an effectively computable constant c depending only on $L=\max \left\{\right|A|,|B|,|C\left|\right\}$ such that if $\left|F\right(A,B,C,u,n\left)\right|=\left|F\right(A,B,C,u,m\left)\right|$ holds for some integers $u,n,m$ with $n\ne m$ then $\left|n\right|,\left|m\right|<c$. For the choices $(A,B,C)\in \left\{\right(0,0,1),(1,-1,1\left)\right\}$ we solve the above equations completely. At the end we give an outlook to the equation $F(0,0,1,u,n)=F(0,0,1,v,m)$ for some fixed integers $n,m$.