Lucas generalized numbers in Narayana's cows sequence

Abstract

Let {N_n}_n≥0 be the Narayana's cows sequence given by N₀ = 0, N₁ = N₂ = 1 and

N_n+3 = N_n+2 + N_n, for integers n ≥ 0

and let {U_n}_{n ≥ 0} be the generalized Lucas sequence with parameters integers a ≥1, b =±1 given by U₀ = 0, U₁ = 1 and

U_n+2 = aU_n+1 + bU_n, for integers n ≥ 0.

In this paper we give effective bounds for the Diophantine equation N_m =U_n, in positive unknowns m and n. We then solve explicitly that equation with Fibonacci, Pell and Balancing sequences cases.