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A228039 Thue-Morse sequence along the squares: A010060(n^2). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0

COMMENTS

(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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

M. Drmota, C. Mauduit, J. Rivat, The Thue-Morse Sequence Along The Squares is Normal, Abstract, ÖMG-DMV Congress, 2013.

A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. 13 (1967/1968) 259-265.

C. Mauduit, J. Rivat, La somme des chiffres des carres, Acta Mathem. 203 (1) (2009) 107-148.

Wikipedia, Normal number

Index entries for characteristic functions

FORMULA

a(n) = A010060(n^2) = A010060(A000290(n)).

MATHEMATICA

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: *)

ThueMorse[Range[0, 104]^2] (* Michael De Vlieger, Dec 22 2017 *)

PROG

(PARI) a(n)=hammingweight(n^2)%2 \\ Charles R Greathouse IV, May 08 2016

CROSSREFS

Cf. A000290, A010060, A293162.

Sequence in context: A280261 A194683 A188295 * A288861 A296211 A284929

Adjacent sequences:  A228036 A228037 A228038 * A228040 A228041 A228042

KEYWORD

nonn,easy

AUTHOR

Jonathan Sondow, Sep 02 2013

STATUS

approved

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Last modified February 17 22:32 EST 2018. Contains 299297 sequences. (Running on oeis4.)