%I A003159 M2306
%S A003159 1,3,4,5,7,9,11,12,13,15,16,17,19,20,21,23,25,27,28,29,31,33,35,36,37,
%T A003159 39,41,43,44,45,47,48,49,51,52,53,55,57,59,60,61,63,64,65,67,68,69,71,
%U A003159 73,75,76,77,79,80,81,83,84,85,87,89,91,92,93,95,97,99,100,101,103,105
%N A003159 Numbers n such that binary representation ends in even number of zeros.
%C A003159 Minimal with respect to property that parity of number of 1's in binary 
               expansion alternates.
%C A003159 Minimal with respect to property that sequence is double of its complement.
%C A003159 If n appears then 2n does not.
%C A003159 Increasing sequence of positive integers n such that A035263(n)=1 (from 
               paper by Allouche et al.). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jan 15 2003
%C A003159 a(n) is an odious number (see A000069) for n odd; a(n) is an evil number 
               (see A001969) for n even. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 16 2004
%C A003159 Indices of odd numbers in A007913, in A001511. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 27 2004
%C A003159 Partial sums of A026465. - Paul Barry (pbarry(AT)wit.ie), Dec 09 2004
%C A003159 A121701(2*a(n)) = A121701(a(n)); A096268(a(n)-1) = 0. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Aug 16 2006
%C A003159 A different permutation of the same terms may be found in A141290 and 
               the accompanying array. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 14 2008
%C A003159 a(n) = n-th clockwise Tower of Hanoi move; counterclockwise if not in 
               the sequence. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 14 2008
%C A003159 Indices of terms of Thue-Morse sequence A010060 which are different from 
               the previous term. [From Tanya Khovanova (tanyakh(AT)yahoo.com), 
               Jan 06 2009]
%D A003159 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003159 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a 
               special sequence, Fib. Quart., 10 (1972), 499-518, 550.
%D A003159 Michael Domaratzki, Trajectory-based codes, Acta Informatica, Volume 
               40, Numbers 6-7 / May, 2004. [From N. J. A. Sloane, Jul 10 2009]
%D A003159 Clark Kimberling, Complementary Equations, Journal of Integer Sequences, 
               Vol. 10 (2007), Article 07.1.4.
%D A003159 Problem E2850, Amer. Math. Monthly, 87 (1980), 671.
%H A003159 T. D. Noe, <a href="b003159.txt">Table of n, a(n) for n=1..1000</a>
%H A003159 J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, 
               Simon Plouffe and Bruce E. Sagan, <a href="http://www-igm.univ-mlv.fr/
               ~berstel/Articles/Relative.ps">A sequence related to that of Thue-Morse</
               a>, Discrete Math., 139 (1995), 455-461.
%H A003159 J.-P. Allouche and J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/
               ~shallit/Papers/ubiq.ps">The Ubiquitous Prouhet-Thue-Morse Sequence</
               a>, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences 
               and Their Applications: Proceedings of SETA '98, Springer-Verlag, 
               1999, pp. 1-16.
%H A003159 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://www.lri.fr/
               ~allouche/kimb.ps">Self-generating sets, integers with missing blocks 
               and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A003159 E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/pdf/math.CO/0407326">
               Congruences for Catalan and Motzkin numbers and related sequences</
               a>, J. Num. Theory 117 (2006), 191-215.
%H A003159 A. S. Fraenkel, <a href="http://www.integers-ejcnt.org/">New games related 
               to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial 
               Number Theory, Vol. 4, Paper G6, 2004.
%H A003159 A. S. Fraenkel, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/">
               Home Page</a>
%H A003159 <a href="Sindx_Bi.html#binary">Index entries for sequences related to 
               binary expansion of n</a>
%F A003159 a(0)=1; for n>=0, a(n+1) = a(n)+1 if (a(n)+1)/2 is not already in the 
               sequence, = a(n)+2 otherwise.
%F A003159 Lim n ->infinity a(n)/n = 3/2 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 13 2002
%F A003159 a(n)=sum(k=1, n, A026465(k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 31 2003
%F A003159 a(n+1) = (if a(n) mod 4 = 3 then A007814(a(n) + 1) mod 2 else a(n) mod 
               2) + a(n) + 1; a(1) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 03 2003
%F A003159 a(A003157(n)) is even. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 
               22 2004
%F A003159 Sequence consists of numbers of the form 4^i*(2j+1), i>=0, j>=0. - Jon 
               Perry (perry(AT)globalnet.co.uk), Jun 06 2004
%F A003159 G.f.: (1/(1-x))Product{k>=1, 1+x^A001045(k)}. - Paul Barry (pbarry(AT)wit.ie), 
               Dec 09 2004
%e A003159 1=1 (odd), 3=11 (even), 4=100 (odd), 5=101 (even), 7=111 (odd), ...
%t A003159 f[n_Integer] := Block[{k = n, c = 0}, While[ EvenQ[k], c++; k /= 2]; 
               c]; Select[ Range[105], EvenQ[ f[ # ]] & ]
%o A003159 (PARI) a(n)=if(n<2,n>0,n=a(n-1); until(valuation(n,2)%2==0,n++); n)
%Y A003159 Indices of even numbers in A007814. Complement of A036554, also one-half 
               of A036554. Cf. A001285, A010060.
%Y A003159 Equals A056196(n)/8.
%Y A003159 Sequence in context: A156246 A136014 A112930 this_sequence A141259 A047501 
               A035242
%Y A003159 Adjacent sequences: A003156 A003157 A003158 this_sequence A003160 A003161 
               A003162
%K A003159 nonn,nice,easy,eigen
%O A003159 1,2
%A A003159 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A003159 Additional comments from Michael Somos.