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Search: id:A001855
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| A001855 |
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Sorting numbers: maximal number of comparisons for sorting n elements by binary insertion.
(Formerly M2433 N0963)
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+0
14
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| 0, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, 33, 37, 41, 45, 49, 54, 59, 64, 69, 74, 79, 84, 89, 94, 99, 104, 109, 114, 119, 124, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Equals n-1 times the expected number of probes for a successful binary search in a size n-1 list.
Piecewise linear: breakpoints at powers of 2 with values given by A000337.
a(n) is the number of digits in the binary representation of all the numbers 1 to n-1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Dec 05 2006
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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 and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.3.1, Eq. (3); Sect. 6.2.1 (4).
J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 48.
Tianxing Tao, On optimal arrangement of 12 points, pp. 229-234 in Combinatorics, Computing and Complexity, ed. D. Du and G. Hu, Kluwer, 1989.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to sorting
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FORMULA
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Let n = 2^(k-1) + g, 0 <= g < 2^(k-1); then a(n) = 1 + n*k - 2^k. - N. J. A. Sloane (njas(AT)research.att.com), Dec 01 2007
a(n) = Sum_{k=1..n} ceiling(log_2 k) = n*ceiling(log_2 n) - 2^ceiling(log_2(n)) + 1.
a(n)=a(floor(n/2))+a(ceiling(n/2))+n-1.
G.f.: x/(1-x)^2 * Sum(k>=0, x^2^k). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 13 2002
a(1)=0, for n>1, a(n)=ceiling(n*a(n-1)/(n-1)+1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 26 2003
a(n) = n-1 + min { a(k)+a(n-k) : 1 <= k <= n-1 }, cf. A003314. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 15 2004
a(n) = A061168(n-1)+n-1 for n>1 - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Dec 05 2006
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MAPLE
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Digits:=200; f:=proc(n) local k, x; x:=floor(log(n)/log(2)); k:=x+1; 1+n*k-2^k; end; - N. J. A. Sloane (njas(AT)research.att.com), Dec 01 2007
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PROGRAM
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(PARI) a(n)=if(n<2, 0, n-1+a(n\2)+a((n+1)\2))
(PARI) a(n)=local(m); if(n<2, 0, m=length(binary(n-1)); n*m-2^m+1)
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CROSSREFS
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Partial sums of A029837.
Cf. A003071, A000337, A030190, A030308, A061168.
Cf. A061168.
Sequence in context: A102696 A130262 A094228 this_sequence A006591 A052488 A076372
Adjacent sequences: A001852 A001853 A001854 this_sequence A001856 A001857 A001858
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KEYWORD
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nonn,easy,nice
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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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Additional comments from M. D. McIlroy (mcilroy(AT)dartmouth.edu).
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