Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers

TL;DR

This paper explores the relationship between Motzkin numbers and Fibonacci numbers by analyzing nine OEIS sequences and introducing the concept of shadows, revealing a connection through sequence reversal.

Contribution

It introduces the concept of Motzkin shadows and demonstrates their link to Fibonacci numbers via sequence reversal, providing new insights into these combinatorial sequences.

Findings

Motzkin shadows reverse to Fibonacci sequences

A connection between Motzkin and Fibonacci numbers is established

Analysis of nine OEIS sequences reveals underlying relationships

Abstract

In this paper, we consider nine OEIS sequences, the analysis of which allows us to find a connection between Motzkin numbers and Fibonacci numbers. In each Motzkin number, we distinguish an even component and an odd component, the difference of these two components is called the shadow of the Motzkin number. Reverse of the Motzkin shadows give us a sequence from the family of Fibonacci numbers.