Initial non-repetitive complexity of infinite words

The initial non-repetitive complexity function   of an infinite word x (first introduced by Moothathu) is the function of nn that counts the number of distinct factors of length nn that appear at the beginning of x prior to the first repetition of a length-nn factor. We examine general properties of the initial non-repetitive complexity function, as well as obtain formulas for the initial non-repetitive complexity of the Thue–Morse word, the Fibonacci word and the Tribonacci word.