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Search: id:A000788
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| A000788 |
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Total number of 1's in binary expansions of 0, ..., n.
(Formerly M0964 N0360)
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+0
23
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| 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, 57, 60, 64, 67, 71, 75, 80, 81, 83, 85, 88, 90, 93, 96, 100, 102, 105, 108, 112, 115, 119, 123, 128, 130, 133, 136, 140, 143, 147, 151, 156, 159, 163, 167, 172, 176, 181, 186
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
J.-P. Allouche & J. Shallit, Automatic sequences, Cambrige University Press, 2003, p. 94
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.
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]
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LINKS
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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
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FORMULA
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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
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CROSSREFS
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The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015.
Cf. A005183.
Adjacent sequences: A000785 A000786 A000787 this_sequence A000789 A000790 A000791
Sequence in context: A140206 A007818 A158618 this_sequence A053039 A027861 A062428
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jan 15 2001
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