1,2

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.

En francais: les nombres impies.

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

Nim-values for game of mock turtles played with n coins.

A115384(n) = number of odious numbers <= n; A000120(a(n))=A132680(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 26 2007

Indices of ones in the Thue-Morse sequence A010060. [From Tanya Khovanova (tanyakh(AT)yahoo.com), Dec 29 2008]

Contribution from Pietro Majer (majer(AT)dm.unipi.it), Mar 15 2009: (Start)

For any positive integer m, the partition of the set of the first 2^m

positive integer numbers into evil ones E and odious ones O is a fair

division for any polynomial sequence p(k) of degree less than m, that is,

sum_{k in E}p(k)=sum_{k in O}p(k) holds for any polynomial p with deg(p)<m

(End)

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 433.

R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.

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.

J. Lambek and L. Moser, On some two way classifications of integers, Canad. Math. Bull. 2 (1959), 85-89.

M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., 3 (1974), 255-261.

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.

D. J. Newman, A Problem Seminar, Springer; see Problem #89.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10001

J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II

J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.

Eric Weisstein's World of Mathematics, Odious Number

Index entries for sequences related to binary expansion of n

Index entries for "core" sequences

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

a(n) = 1/2 * (4n + 1 + (-1)^A000120(n)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 14 2003

Numbers n such that A010060(n)=1 - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 15 2003

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

(-1)^a(n)=2*A010060(n)-1 - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 08 2004

a(0) = 1, a(2n) = a(n) + 2n, a(2n+1) = -a(n) + 6n + 3.

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.

Select[Range[300], OddQ[DigitCount[ #, 2][[1]]] &] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 31 2006

(PARI) a(n)=2*n+1-subst(Pol(binary(n)), x, 1)%2

(PARI) a(n)=if(n<1, 1, if(n%2==0, a(n/2)+n, -a((n-1)/2)+3*n))

The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015.

Complement of A001969 (the evil numbers). Cf. A133009.

a(n)=2*n+1-A010060(n)=A001969(n)+(-1)^A010060(n).

First differences give A007413.

Cf. A000773.

Note that A000079, A083420, A002042, A002089, A132679 are subsequences.

A019568 [From Pietro Majer (majer(AT)dm.unipi.it), Mar 15 2009]

Sequence in context: A050082 A112648 A161989 this_sequence A140137 A080308 A089559

Adjacent sequences: A000066 A000067 A000068 this_sequence A000070 A000071 A000072

easy,core,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).