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 $58425=\mathcal{ N}_{ 31}=\overline{ 3332200}_5=\overline{ 332223}_7.$
Categories: math.NT