The generalized order-k Narayana's cows numbers

Abstract

Narayana's cows problem is a problem similar to the Fibonacci's rabbit problem. We define the numbers which are the solutions of this problem as Narayana's cows numbers. Narayana's cows sequence satisfies the third order recurrence relation Nr = Nr-1+Nr-3 with initial condition N0 = N1 = N2 = 1. In this paper, we define a new recurrence which is called generalized order-k Narayana's cows sequence and by using this generalization and some matrix properties, we give some identities related to the Narayana's cows numbers.