Counting Colorful Necklaces and Bracelets in Three Colors

A necklace or bracelet is $\mathit{\text{colorful}}$ if no pair of adjacent beads are the same color. In addition, two necklaces are $\mathit{\text{equivalent}}$ if one results from the other by permuting its colors, and two bracelets are $\mathit{\text{equivalent}}$ 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 $K^{\mathrm{\prime }}(n)$ of colorful bracelets formed with $n$-beads in at most three colors. Expressions obtained for $K^{\mathrm{\prime }}(n)$ simplify expressions given by OEIS sequence A114438, while the expressions given for $K(n)$ appear to be new and are not included in OEIS.