login
a(n) = floor(S2(n)/2) mod 2, where S2(n) is the binary weight of n.
3

%I #15 Sep 28 2016 14:57:25

%S 0,0,0,1,0,1,1,1,0,1,1,1,1,1,1,0,0,1,1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,1,

%T 1,1,1,1,1,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,1,0,1,1,1,

%U 1,1,1,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,0,0

%N a(n) = floor(S2(n)/2) mod 2, where S2(n) is the binary weight of n.

%C A generalized Thue Morse sequence.

%C A class of generalized Thue-Morse sequences: Let F(t) be an integer function, m,k integers. Let Sk(n) be sum of digits of n; n in base-k. Then a(n)= F(Sk(n)) mod m is a generalized Thue-Morse sequence. Thue-Morse sequence has F(t)=t (identity function), S2(n), m=2,k=2. Interesting properties have sequences where F(Sk(n))=floor(Q*Sk(n)); Q is a positive rational number; a(n)=floor(Q*Sk(n)) mod m. Another interesting sequences are a(n)=(n*Sk(n)) mod m; a(n)=(n+Sk(n)) mod m.

%D J. P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, 2003.

%H G. C. Greubel, <a href="/A135136/b135136.txt">Table of n, a(n) for n = 0..10000</a>

%H Ricardo Astudillo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Astudillo/astudillo12.html">On a class of Thue-Morse type sequences</a>, Journal of Integer Sequences, Vol. 6 (2003), Article 03.4.2

%H R. Bacher and R. Chapman, <a href="http://dx.doi.org/10.1016/j.ejc.2003.06.001">Symmetric Pascal matrices modulo p</a>, European J. Combin. 25 (2004), 459-473.

%t Table[Mod[Floor[(Plus @@ IntegerDigits[n, 2])/2], 2], {n, 0, 90}] (* _Stefan Steinerberger_, Feb 14 2008 *)

%Y Cf. A010060.

%K nonn

%O 0,1

%A _Ctibor O. Zizka_, Feb 13 2008

%E More terms from _Stefan Steinerberger_, Feb 14 2008