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 $\rho\geqslant2$ 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 $1 \leqslant \rho \leqslant 10$ is $58425=\mathcal{N}_{31}=\overline{3332200}_5=\overline{332223}_7.$