Search: id:A001969
Results 1-1 of 1 results found.

%I A001969 M2395 N0952
%S A001969 0,3,5,6,9,10,12,15,17,18,20,23,24,27,29,30,33,34,36,39,40,43,45,46,
%T A001969 48,51,53,54,57,58,60,63,65,66,68,71,72,75,77,78,80,83,85,86,89,90,92,
%U A001969 95,96,99,101,102,105,106,108,111,113,114,116,119,120,123,125,126,129
%N A001969 Evil numbers: numbers with an even number of 1's in their binary expansion.
%C A001969 This sequence and A000069 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 A001969 En francais: les nombres paiens.
%C A001969 a(n)-A001285(n) = 2n-1 has been verified for n=0,1,2,...,400 - John W. 
               Layman (layman(AT)math.vt.edu)
%C A001969 First differences give A036585. Observed by Franklin T. Adams-Watters, 
               proved by Max Alekseyev, Aug 30 2006. This is equivalent to a(n) 
               = 2*n + A010060(n). Proof: If the number of bits in n is odd then 
               the last bit of a(n) is 1 and if the number of bits in n is even 
               then the last bit of a(n) is 0. Hence the sequence of last bits is 
               A010060. Therefore a(n) = 2*n + A010060(n).
%C A001969 Indices of zeros in the Thue-Morse sequence A010060. [From Tanya Khovanova 
               (tanyakh(AT)yahoo.com), Feb 13 2009]
%D A001969 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001969 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001969 Allouche, Jean-Paul and Cohen, Henri, Dirichlet series and curious infinite 
               products, Bull. London Math. Soc. 17 (1985), 531-538.
%D A001969 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical 
               Computer Sci., 98 (1992), 163-197.
%D A001969 J. Lambek and L. Moser, On some two way classifications of integers, 
               Canad. Math. Bull. 2 (1959), 85-89.
%D A001969 M. D. McIlroy, The number of 1's in binary integers: bounds and extremal 
               properties, SIAM J. Comput., 3 (1974), 255-261.
%D A001969 H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number 
               Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
%D A001969 D. J. Newman, A Problem Seminar, Springer; see Problem #89.
%D A001969 Shallit, J. O., On infinite products associated with sums of digits, 
               J. Number Theory 21 (1985), 128-134.
%H A001969 N. J. A. Sloane, <a href="b001969.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A001969 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/
               Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer 
               Sci., 98 (1992), 163-197.
%H A001969 J. N. Cooper, D. Eichhorn and K. O'Bryant, <a href="http://arXiv.org/
               abs/math.NT/0506496">Reciprocals of binary power series</a>
%H A001969 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EvilNumber.html">Evil Number</a>
%H A001969 <a href="Sindx_Bi.html#binary">Index entries for sequences related to 
               binary expansion of n</a>
%H A001969 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001969 Note that 2n+1 is in the sequence iff 2n is not and so this sequence 
               has asymptotic density 1/2. - Franklin T. Adams-Watters, Aug 23 2006
%F A001969 a(n) = (1/2) * (4n + 1 - (-1)^A000120(n)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Sep 14 2003
%F A001969 G.f.: sum[k>=0, t(3+2t+3t^2)/(1-t^2)^2 * prod(l=0, k-1, 1-x^(2^l)), t=x^2^k]. 
               - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 25 2004
%F A001969 n such that A010060(n)=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Nov 15 2003
%F A001969 a(2*n+1) + a(2*n) = A017101(n) = 8*n+3. a(2*n+1) - a(2*n) gives the Thue-Morse 
               sequence (3, 1 version): 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, .... A001969(n) 
               + A000069(n) = A016813(n) = 4*n+1. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Feb 04 2004
%F A001969 a(0) = 0, a(2n) = a(n) + 2n, a(2n+1) = -a(n) + 6n + 3.
%F A001969 Let b(n) = 1 if sum of digits of n is even, -1 if it is odd; then Shallit 
               (1985) showed that Prod_{n>=0} ((2n+1)/(2n+2))^b(n) = 1/sqrt(2).
%F A001969 a(n) = 2n + A010060(n). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Aug 28 2006
%p A001969 s := proc(n) local i,j,ans; ans := [ ]; j := 0; for i from 0 while j<n 
               do if add(k,k=convert(i,base,2)) mod 2=0 then ans := [ op(ans),i 
               ]; j := j+1; fi; od; RETURN(ans); end; t1 := s(100); A001969 := n->
               t1[n]; # s(k) gives first k terms.
%t A001969 Select[Range[0,300], EvenQ[DigitCount[ #, 2][[1]]] &]
%o A001969 (PARI) a(n)=2*n+subst(Pol(binary(n)),x,1)%2
%o A001969 (PARI) a(n)=if(n<1,0,if(n%2==0,a(n/2)+n,-a((n-1)/2)+3*n))
%o A001969 (MAGMA) [ n : n in [0..129] | IsEven(&+Intseq(n,2)) ]; - from Sergei 
               Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%Y A001969 Complement of A000069 (the odious numbers). Cf. A133009.
%Y A001969 a(n)=2*n+A010060(n)=A000069(n)-(-1)^A010060(n). Cf. A018900.
%Y A001969 The basic sequences concerning the binary expansion of n are A000120, 
               A000788, A000069, A001969, A023416, A059015.
%Y A001969 Cf. A036585 (differences), A010060.
%Y A001969 Sequence in context: A081121 A165740 A080307 this_sequence A075311 A032786 
               A080309
%Y A001969 Adjacent sequences: A001966 A001967 A001968 this_sequence A001970 A001971 
               A001972
%K A001969 easy,core,nonn,nice
%O A001969 1,2
%A A001969 N. J. A. Sloane (njas(AT)research.att.com).
%E A001969 More terms from Robin Trew (trew(AT)hcs.harvard.edu).

Search completed in 0.002 seconds