  Narayana's cows sequence satisfies the third-order linear recurrence relation
$N_n=N_{n-1}+N_{n-3}$ for $n \geq 3$ with initial conditions $N_0=0$ and
$N_1=N_2=1$. In this paper, we study $b$-repdigits which are sums of two
Narayana numbers. We explicitly determine these numbers for the bases $2\le
b\leq100$ as an illustration. We also obtain results on the existence of
Mersenne prime numbers, 10-repdigits, and numbers with distinct blocks of
digits in the Narayana sequence. The proof of our main theorem uses lower
bounds for linear forms in logarithms and a version of the Baker-Davenport
reduction method in Diophantine approximation.
