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A228039
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Thue-Morse sequence along the squares:A010060(n^2).
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4
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0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0
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OFFSET
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0
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COMMENTS
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(Adapted from Drmota, Mauduit, and Rivat) The Thue-Morse sequence T(n) is a 0-1-sequence that can be defined by T(n) = s2(n) mod 2, where s2(n) denotes the binary sum-of-digits function of n (that is, the number of powers of 2). By definition it is clear that 0 and 1 appear with the same asymptotic frequency 1/2. However, there is no consecutive block of the form 000 or 111, so that the Thue-Morse sequence is not normal. (A 0-1-sequence is normal if every finite 0-1-block appears with the asymptotic frequency 1/2^k, where k denotes the length of the block.) Mauduit and Rivat (2009) showed that the subsequence T(n^2) also has the property that both 0 and 1 appear with the same asymptotic frequency 1/2. This solved a long-standing conjecture by Gelfond (1967/1968). Drmota, Mauduit, and Rivat (2013) proved that the subsequence T(n^2) is actually normal.
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LINKS
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FORMULA
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MATHEMATICA
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a[n_]:= If[ n == 0, 0, If[ Mod[n, 2] == 0, a[n/2], 1 - a[(n - 1)/2]]]; Table[ a[n^2], {n, 0, 104}]
(* Second program: *)
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PROG
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(Python)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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