Identities involving Narayana numbers

TL;DR

This paper explores identities involving Narayana's cows numbers, a sequence defined by a third-order recurrence, and expresses certain sub-sequences in terms of other sequence elements, including sums of four-step apart terms.

Contribution

It derives new identities linking Narayana's cows numbers with shifted and summed terms, expanding understanding of their recursive structure.

Findings

Expressed $ar+b$ subscripted Narayana numbers using three $a$ step apart numbers.

Derived formulas for sums of four-step apart Narayana numbers.

Established recurrence relations for Narayana's cows sequence.

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 $N_{r}=N_{r-1}+N_{r-3}$ ($r\geq3$) with initial condition $N_{0} =0$, $N_{1} = N_{2}= 1$. In this paper, the $ar+b$ subscripted Narayana numbers will be expressed by three $a$ step apart Narayana numbers for any $1\leq b\leq a$ ($a\in \mathbb{Z}$). Furthermore, we study the sum $S_{N,r}^{(4,b)}=\sum_{k=0}^{r}N_{4k+b}$ of $4$ step apart Narayana numbers for any $1\leq b\leq 4$.