Counting colorful necklaces and bracelets in three colors

摘要

A necklace or bracelet is colorful if no pair of adjacent beads are the same color. In addition, two necklaces are equivalent if one results from the other by permuting its colors, and two bracelets are 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 similar to (n) of colorful bracelets formed with n-beads in at most three colors. Expressions obtained for K similar to(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.