Lucas generalized numbers in Narayana's cows sequence

Abstract

Let {Nn}n≥0 be the Narayana's cows sequence given by N0 = 0, N1 = N2 = 1 and Nn+3 = Nn+2 + Nn, for integers n ≥ 0 and let {Un}n ≥ 0 be the generalized Lucas sequence with parameters integers a ≥1, b =±1 given by U0 = 0, U1 = 1 and Un+2 = aUn+1 + bUn, for integers n ≥ 0. In this paper we give effective bounds for the Diophantine equation Nm =Un, in positive unknowns m and n. We then solve explicitly that equation with Fibonacci, Pell and Balancing sequences cases.