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%I A000069 M1031 N0388
%S A000069 1,2,4,7,8,11,13,14,16,19,21,22,25,26,28,31,32,35,37,38,41,42,44,47,
%T A000069 49,50,52,55,56,59,61,62,64,67,69,70,73,74,76,79,81,82,84,87,88,91,93,
%U A000069 94,97,98,100,103,104,107,109,110,112,115,117,118,121,122,124,127,128
%N A000069 Odious numbers: numbers with an odd number of 1's in their binary expansion.
%C A000069 This sequence and A001969 give the unique solution to the problem of 
               splitting the nonnegative integers into two classes in such a way 
               that sums of pairs of distinct elements from either class occur with 
               the same multiplicities [Lambek and Moser]. Cf. A000028, A000379.
%C A000069 En francais: les nombres impies.
%C A000069 Has asymptotic density 1/2, since exactly 2 of the 4 numbers 4k, 4k+1, 
               4k+2, 4k+3 have an even sum of bits, while the other 2 have an odd 
               sum. - J. O. Shallit, Jun 04, 2002
%C A000069 Nim-values for game of mock turtles played with n coins.
%C A000069 A115384(n) = number of odious numbers <= n; A000120(a(n))=A132680(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 26 2007
%C A000069 Indices of ones in the Thue-Morse sequence A010060. [From Tanya Khovanova 
               (tanyakh(AT)yahoo.com), Dec 29 2008]
%C A000069 Contribution from Pietro Majer (majer(AT)dm.unipi.it), Mar 15 2009: (Start)
%C A000069 For any positive integer m, the partition of the set of the first 2^m
%C A000069 positive integer numbers into evil ones E and odious ones O is a fair
%C A000069 division for any polynomial sequence p(k) of degree less than m, that 
               is,
%C A000069 sum_{k in E}p(k)=sum_{k in O}p(k) holds for any polynomial p with deg(p)<m
%C A000069 (End)
%D A000069 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. 
               Computer Sci., 307 (2003), 3-29.
%D A000069 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, 
               NY, 2 vols., 1982, see p. 433.
%D A000069 R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. 
               K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.
%D A000069 R. K. Guy, The unity of combinatorics, Proc. 25th Iranian Math. Conf, 
               Tehran, (1994), Math. Appl 329 129-159, Kluwer Dordrecht 1995, Math. 
               Rev. 96k:05001.
%D A000069 J. Lambek and L. Moser, On some two way classifications of integers, 
               Canad. Math. Bull. 2 (1959), 85-89.
%D A000069 M. D. McIlroy, The number of 1's in binary integers: bounds and extremal 
               properties, SIAM J. Comput., 3 (1974), 255-261.
%D A000069 H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number 
               Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
%D A000069 D. J. Newman, A Problem Seminar, Springer; see Problem #89.
%D A000069 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.
%D A000069 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000069 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000069 N. J. A. Sloane, <a href="b000069.txt">Table of n, a(n) for n = 1..10001</
               a>
%H A000069 J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">
               The Ring of k-regular Sequences, II</a>
%H A000069 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 A000069 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               OdiousNumber.html">Odious Number</a>
%H A000069 <a href="Sindx_Bi.html#binary">Index entries for sequences related to 
               binary expansion of n</a>
%H A000069 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000069 G.f.: 1+sum[k>=0, t(2+2t+5t^2-t^4)/(1-t^2)^2 * prod(l=0, k-1, 1-x^(2^l)), 
               t=x^2^k]. - Ralf Stephan, Mar 25 2004
%F A000069 a(n) = 1/2 * (4n + 1 + (-1)^A000120(n)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Sep 14 2003
%F A000069 Numbers n such that A010060(n)=1 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Nov 15 2003
%F A000069 a(2*n+1) + a(2*n) = A017101(n) = 8*n+3 . a(2*n+1) - a(2*n) gives the 
               Thue-Morse sequence (1, 3 version): 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 
               1, 3, 1, ... A001969(n) + A000069(n) = A016813(n) = 4*n+1 . - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), Feb 04 2004
%F A000069 (-1)^a(n)=2*A010060(n)-1 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Mar 08 2004
%F A000069 a(0) = 1, a(2n) = a(n) + 2n, a(2n+1) = -a(n) + 6n + 3.
%p A000069 s := proc(n) local i,j,k,b,sum,ans; ans := [ ]; j := 0; for i while j<n 
               do sum := 0; b := convert(i,base,2); for k to nops(b) do sum := sum+b[ 
               k ]; od; if sum mod 2 = 1 then ans := [ op(ans),i ]; j := j+1; fi; 
               od; RETURN(ans); end; t1 := s(100); A000069 := n->t1[n]; # s(k) gives 
               first k terms.
%t A000069 Select[Range[300], OddQ[DigitCount[ #, 2][[1]]] &] - Stefan Steinerberger 
               (stefan.steinerberger(AT)gmail.com), Mar 31 2006
%o A000069 (PARI) a(n)=2*n+1-subst(Pol(binary(n)),x,1)%2
%o A000069 (PARI) a(n)=if(n<1,1,if(n%2==0,a(n/2)+n,-a((n-1)/2)+3*n))
%Y A000069 The basic sequences concerning the binary expansion of n are A000120, 
               A000788, A000069, A001969, A023416, A059015.
%Y A000069 Complement of A001969 (the evil numbers). Cf. A133009.
%Y A000069 a(n)=2*n+1-A010060(n)=A001969(n)+(-1)^A010060(n).
%Y A000069 First differences give A007413.
%Y A000069 Cf. A000773.
%Y A000069 Note that A000079, A083420, A002042, A002089, A132679 are subsequences.
%Y A000069 A019568 [From Pietro Majer (majer(AT)dm.unipi.it), Mar 15 2009]
%Y A000069 Sequence in context: A050082 A112648 A161989 this_sequence A140137 A080308 
               A089559
%Y A000069 Adjacent sequences: A000066 A000067 A000068 this_sequence A000070 A000071 
               A000072
%K A000069 easy,core,nonn,nice
%O A000069 1,2
%A A000069 N. J. A. Sloane (njas(AT)research.att.com).

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