# Enumeration of Restricted Words and Linear Recurrence Equations

## TL;DR

This paper explores the enumeration of restricted words over finite alphabets, deriving explicit formulas for related functions and connecting these to well-known sequences like Fibonacci and Tribonacci, while also solving linear recurrence equations.

## Contribution

It introduces a reverse approach to enumerate restricted words by identifying initial functions that generate known sequences and derives explicit formulas for these functions and related recurrence solutions.

## Key findings

- Explicit formulas for functions $f_m$ and $c_m$ for five types of restricted words.
- Connections between these functions and classical sequences like Fibonacci, Mersenne, Pell, etc.
- Combinatorial derivations of solutions to linear recurrence equations.

## Abstract

In previous papers, for an arithmetical function $f_0$, we defined functions $f_m$ and $c_m$ and designated numbers of restricted words over a finite alphabet counted by these functions. In this paper, we examine the reverse problem for five specific types of restricted words. Namely, we find the initial function $f_0$ such that $f_m$ and $c_m$ enumerate these words. In each case, we derive explicit formulas for $f_m$ and $c_m$. Fibonacci, Merssen, Pell, Jacosthal, Tribonacci, and Padovan numbers all appear as values of $f_m$, so we obtain new formulas for these numbers. Also, we combinatorially derive explicit formulas for the solutions of five types of homogenous linear recurrence equations.