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  "body": "\n## Description\n\nThe *Thue–Morse sequence* is the binary [[sequence|0G9EKM0E15]] $(t_n)_{n \\ge 0}$ given by $t_0 = 0$, $t_{2n} = t_n$ and $t_{2n+1} = 1 - t_n$.\nEquivalently, $t_n$ is $0$ if the binary expansion of $n$ has an even number of digits $1$, and $1$ otherwise.\nIts first sixteen terms are\n$$0110\\,1001\\,1001\\,0110.$$\n\n## Remarks\n\nCheck: $t_5 = 0$ because $5 = 101_2$ has two digits $1$, and $t_7 = 1$ because $7 = 111_2$ has three.\n\nThe sequence is the fixed point, starting with $0$, of the morphism $\\mu$ of words with $\\mu(0) = 01$ and $\\mu(1) = 10$.\nSo the first $2^{m+1}$ terms are the first $2^m$ terms followed by their complement, where $0$ and $1$ are swapped.\n\nThe sequence is overlap-free: it has no factor of the form $a x a x a$, where $a$ is a letter and $x$ is a word.\nIn particular it has no factor of the form $u u u$ with $u$ nonempty.\nThue proved overlap-freeness, and it makes the sequence a basic example in [[combinatorics on words|YJ961M051W]].\nIt is not periodic, and it is a $2$-automatic sequence, computed by a finite automaton that reads $n$ in base $2$.\n\n## Explanations\n\n### L1 human\n\nStart with $0$, and at each stage append the complement of what you have.\nThis gives $0$, then $01$, then $0110$, then $01101001$, and so on.\nThe result never repeats any block three times in a row.\nIt is also used as a fair order of turns: two players who pick in the order $A B B A B A A B \\ldots$ share the advantage of going first.\n",
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