0,3

J.-P. Allouche & J. Shallit, Automatic sequences, Cambrige University Press, 2003, p. 94

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]

E. N. Gilbert, Games of identification or convergence, SIAM Review, 4 (1962), 16-24.

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.

Z. Li and E. M. Reingold, Solution of a divide-and-conquer maximin recurrence, SIAM J. Comput., 18 (1989), 1188-1200.

B. Lindstrom, On a combinatorial problem in number theory, Canad. Math. Bull., 8 (1965), 477-490.

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).

T. D. Noe, Table of n, a(n) for n=0..1000

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

P. J. Grabner, H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Eric Weisstein's World of Mathematics, Binary

Index entries for sequences related to binary expansion of n

a(n)=sum(k=1, n, A000120(k)). - Benoit Cloitre, Dec 19, 2002

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

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

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

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

A000788(2^n-1) = A001787(n) = n*2^(n-1). [From M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 22 2009]

Contribution from M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 22 2009: (Start)

(PARI) A000788(n)={ n<3 & return(n); if( bittest(n, 0) \\

, n+1 == 1<<valuation(n+1, 2) && return(valuation(n+1, 2)*(n+1)/2) \\

; A000788(n>>1)*2+n>>1+1 \\

, n == 1<<valuation(n, 2) && return(valuation(n, 2)*n/2+1) \\

; A000788(n>>=1)+A000788(n-1)+n )} \\ (End)

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

Cf. A005183.

Sequence in context: A140206 A007818 A158618 this_sequence A053039 A027861 A062428

Adjacent sequences: A000785 A000786 A000787 this_sequence A000789 A000790 A000791

nonn,nice,easy,new

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

More terms from Larry Reeves (larryr(AT)acm.org), Jan 15 2001