In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet [fr], Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence.
The Prouhet–Thue–Morse constant can also be expressed, without using tn , as an infinite product,[1] This formula is obtained by substituting x = 1/2 into generating series for tn The continued fraction expansion of the constant is [0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …] (sequence A014572 in the OEIS) Yann Bugeaud and Martine Queffélec showed that infinitely many partial quotients of this continued fraction are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.
[2] The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.
Yann Bugaeud proved that the Prouhet–Thue–Morse constant has an irrationality measure of 2.
If a language L over {0, 1} is chosen at random, by flipping a fair coin to decide whether each word w is in L, the probability that it contains at least one word for each possible length is [5]