On k-Pell numbers which are sum of two Narayana's cows numbers

For any positive integer k≥2, let (Pₙ⁽ᵏ⁾)n≥2-k be the k-generalized Pell sequence which starts with 0,⋯,0,1 (k terms) with the linear recurrence P_n⁽ᵏ⁾ = 2Pₙ₋₁⁽ᵏ⁾+Pₙ₋₂⁽ᵏ⁾+⋯+Pₙ₋ₖ⁽ᵏ⁾ n≥2. Let (Nₙ)n≥0 be Narayana's sequence given by N_0=N_1=N_2=1 Nₙ₊₃=Nₙ₊₂+N_n. The purpose of this paper is to determine all k-Pell numbers which are sums of two Narayana's numbers. More precisely, we study the Diophantine equation P_p⁽ᵏ⁾=N_n+N_m in nonnegative integers k, p, n and m.