Repetitions in the infinite n-bonacci word

Abstract The Fibonacci word Fm is defined as the concatenation of the preceding two Fibonacci words Fm−1 and Fm−2. We generalize it to the n-bonacci word as the concatenation of the preceding n words. We consider the structure of the subword repetitions in the n-bonacci word. In Fibonacci word (n = 2), the maximal repetition of the subword is known to be less than 2 + φ, where φ = (1 + √ 5)/2 is the golden ratio. We give a general result for any n >= 2. We prove that the maximal repetition of the finite k-th n-bonacci word in the infinite n-bonacci word converges to 2 + 1/(φ(n) − 1) as k → ∞, and it approaches to 3 as n → ∞, where φ(n) is the n-bonacci constant and φ(2) = φ.