Narayana's cows numbers which are concatenations of three repdigits in base $\rho$

Abstract

Narayana's sequence is a ternary recurrent sequence defined by the recurrence relation ${\mathcal{N}}_n={\mathcal{N}}_{n-1}+{\mathcal{N}}_{n-3}$ with initial terms ${\mathcal{N}}_0=0$ and ${\mathcal{N}}_1={\mathcal{N}}_2={\mathcal{N}}_3=1$. Let $\rho ⩾2$ be a positive integer. In this study, it is proved that the $n$th Narayana's number ${\mathcal{N}}_n$ which is concatenations of three repdigits in base $\rho$ satisfies $n<5.6\cdot 10^{48}\cdot {\log }^7\rho$. Moreover, it is shown that the largest Narayana's number which is concatenations of three repdigits in base $\rho$ with $1⩽\rho ⩽10$ is $58425={\mathcal{N}}_{31}={\overline{3332200}}_5={\overline{332223}}_7.$