%I A000788 M0964 N0360
%S A000788 0,1,2,4,5,7,9,12,13,15,17,20,22,25,28,32,33,35,37,40,42,45,48,52,54,
%T A000788 57,60,64,67,71,75,80,81,83,85,88,90,93,96,100,102,105,108,112,115,119,
%U A000788 123,128,130,133,136,140,143,147,151,156,159,163,167,172,176,181,186
%N A000788 Total number of 1's in binary expansions of 0, ..., n.
%D A000788 J.-P. Allouche & J. Shallit, Automatic sequences, Cambrige University 
               Press, 2003, p. 94
%D A000788 R. Bellman and H. N. Shapiro, On a problem in additive number theory, 
               Annals Math., 49 (1948), 333-340. See Eq. 1.9. [From N. J. A. Sloane, 
               Mar 12 2009]
%D A000788 E. N. Gilbert, Games of identification or convergence, SIAM Review, 4 
               (1962), 16-24.
%D A000788 R. L. Graham, On primitive graphs and optimal vertex assignments, pp. 
               170-186 of Internat. Conf. Combin. Math. (New York, 1970), Annals 
               of the NY Academy of Sciences, Vol. 175, 1970.
%D A000788 Z. Li and E. M. Reingold, Solution of a divide-and-conquer maximin recurrence, 
               SIAM J. Comput., 18 (1989), 1188-1200.
%D A000788 B. Lindstrom, On a combinatorial problem in number theory, Canad. Math. 
               Bull., 8 (1965), 477-490.
%D A000788 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000788 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000788 T. D. Noe, <a href="b000788.txt">Table of n, a(n) for n=0..1000</a>
%H A000788 S. R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">
               Odd Entries in Pascal's Trinomial Triangle</a> (arXiv:0802.2654)
%H A000788 P. J. Grabner, H.-K. Hwang, <a href="http://algo.stat.sinica.edu.tw/">
               Digital sums and divide-and-conquer recurrences: Fourier expansions 
               and absolute convergence</a>
%H A000788 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A000788 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A000788 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Binary.html">Binary</a>
%H A000788 <a href="Sindx_Bi.html#binary">Index entries for sequences related to 
               binary expansion of n</a>
%F A000788 a(n)=sum(k=1, n, A000120(k)). - Benoit Cloitre, Dec 19, 2002
%F A000788 a(0) = 0, a(2n) = a(n)+a(n-1)+n, a(2n+1) = 2a(n)+n+1. - Ralf Stephan 
               (ralf(AT)ark.in-berlin.de), Sep 13 2003
%F A000788 a(n)=(1/2)*log2(n)*n + O(n); a(2^n)=n*2^(n-1)+1 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Sep 25 2003
%F A000788 a(n)=(1/2)*n*log(n)/log(2)+n*F(log(n)/log(2)) where F is a nowhere differentiable 
               continuous function of period 1 (see Allouche & Shallit) - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Jun 08 2004
%F A000788 G.f.: 1/(1-x)^2 * Sum(k>=0, x^2^k/(1+x^2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Apr 19 2003
%F A000788 A000788(2^n-1) = A001787(n) = n*2^(n-1). [From M. F. Hasler (mhasler(AT)univ-ag.fr), 
               Nov 22 2009]
%o A000788 Contribution from M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 22 2009: 
               (Start)
%o A000788 (PARI) A000788(n)={ n<3 & return(n); if( bittest(n,0) \\
%o A000788 , n+1 == 1<<valuation(n+1,2) && return(valuation(n+1,2)*(n+1)/2) \\
%o A000788 ; A000788(n>>1)*2+n>>1+1 \\
%o A000788 , n == 1<<valuation(n,2) && return(valuation(n,2)*n/2+1) \\
%o A000788 ; A000788(n>>=1)+A000788(n-1)+n )} \\ (End)
%Y A000788 The basic sequences concerning the binary expansion of n are A000120, 
               A000788, A000069, A001969, A023416, A059015.
%Y A000788 Cf. A005183.
%Y A000788 Sequence in context: A140206 A007818 A158618 this_sequence A053039 A027861 
               A062428
%Y A000788 Adjacent sequences: A000785 A000786 A000787 this_sequence A000789 A000790 
               A000791
%K A000788 nonn,nice,easy,new
%O A000788 0,3
%A A000788 N. J. A. Sloane (njas(AT)research.att.com).
%E A000788 More terms from Larry Reeves (larryr(AT)acm.org), Jan 15 2001