|
%I #58 Mar 27 2023 07:51:28
%S 0,1,2,2,3,3,3,4,5,5,5,6,6,7,8,8,9,9,9,10,10,11,12,12,12,13,14,14,15,
%T 15,15,16,17,17,17,18,18,19,20,20,20,21,22,22,23,23,23,24,24,25,26,26,
%U 27,27,27,28,29,29,29,30,30,31,32,32,33,33,33,34,34,35,36,36,36,37,38,38
%N Partial sums of Thue-Morse numbers A010060.
%H G. C. Greubel, <a href="/A115384/b115384.txt">Table of n, a(n) for n = 0..10000</a>
%H Hassan Tarfaoui, <a href="http://d.tarfaoui.free.fr/cg/1990/EX1/exobis.pdf">Concours Général 1990 - Exercice 1</a> (in French).
%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.
%F a(n) = Sum_{k=0..n} A010060(k)^2.
%F a(n+1) = A115382(2n, n).
%F a(n)/n -> 1/2; a(n) = number of odious numbers <= n, see A000069. - _Reinhard Zumkeller_, Aug 26 2007, corrected by _M. F. Hasler_, May 22 2017.
%F a(n) = Sum_{i=1..n} (S2(n) mod 2), where S2 = binary weight; lim a(n)/n = 1/2. More generally, consider a(n) = Sum_{i=1..n} (F(Sk(n)) mod m), where Sk(n) is sum of digits of n, n in base k; F(t) is an arithmetic function; m integer. How does lim a(n)/n depend on F(t)? - _Ctibor O. Zizka_, Feb 25 2008
%F a(n) = n + 1 - A159481(n). - _Reinhard Zumkeller_, Apr 16 2009
%F a(n) = floor((n+1)/2)+(1+(-1)^n)*(1-(-1)^A000120(n))/4. - _Vladimir Shevelev_, May 27 2009
%F G.f.: (1/(1 - x)^2 - Product_{k>=1} (1 - x^(2^k)))/2. - _Ilya Gutkovskiy_, Apr 03 2019
%F a(n) = A026430(n+1) - n - 1. - _Michel Dekking_, Sep 17 2019
%F a(2n+1) = n+1 (see Hassan Tarfaoui link, Concours Général 1990). - _Bernard Schott_, Jan 21 2022
%t Accumulate[Nest[Flatten[#/.{0->{0,1},1->{1,0}}]&,{0},7]] (* _Peter J. C. Moses_, Apr 15 2013 *)
%t Accumulate[ThueMorse[Range[0,100]]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 02 2017 *)
%o (PARI) a(n)=n\2 + (n%2 || hammingweight(n+1)%2==0) \\ _Charles R Greathouse IV_, Mar 22 2013
%o (Python)
%o def A115384(n): return (n>>1)+(n&1|((n+1).bit_count()&1^1)) # _Chai Wah Wu_, Mar 01 2023
%Y Cf. A000069, A000120, A010060, A026430, A115382, A159481.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Jan 21 2006
%E Edited by _M. F. Hasler_, May 22 2017
|
