# Counting Colorful Necklaces and Bracelets in Three Colors

## TL;DR

This paper derives formulas for counting non-equivalent colorful necklaces and bracelets with n beads in up to three colors, extending known sequences and providing new results for necklace counts.

## Contribution

It introduces new formulas for counting colorful necklaces and bracelets under symmetry, with novel expressions for necklaces not previously documented.

## Key findings

- Derived formulas for K(n) and K'(n)
- Simplified expressions for bracelets matching OEIS sequence A114438
- New formulas for counting necklaces not in OEIS

## Abstract

A necklace or bracelet is \textit{colorful} if no pair of adjacent beads are the same color. In addition, two necklaces are \textit{equivalent} if one results from the other by permuting its colors, and two bracelets are \textit{equivalent} if one results from the other by either permuting its colors or reversing the order of the beads; a bracelet is thus a necklace that can be turned over. This note counts the number $K(n)$ of non-equivalent colorful necklaces and the number $K'(n)$ of colorful bracelets formed with $n$-beads in at most three colors. Expressions obtained for $K'(n)$ simplify expressions given by OEIS sequence A114438, while the expressions given for $K(n)$ appear to be new and are not included in OEIS.