The Circular Version of Fibonacci-like Sequences

In this paper, we study how the enumeration of certain strings on a line is related to the enumeration of similar strings on a circle. Using symbols from a finite language, we prove a possibly new equation between the number of circular strings of a certain length that avoid a single pattern and numbers of linear strings of various lengths that avoid the same pattern. Index Terms - Autocorrelation of a Pattern, Avoiding a Pattern, Cyclic String, Generalized Fibonacci Sequence, Generalized Loucas Sequence, Generating Function.