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 A001855 Sorting numbers: maximal number of comparisons for sorting n elements by binary insertion. (Formerly M2433 N0963) 21
 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; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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, Dec 05 2006 It is also coincidentally the maximum number of comparisons for merge sort. - Li-yao Xia, Nov 18 2015 REFERENCES 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. 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). Tianxing Tao, On optimal arrangement of 12 points, pp. 229-234 in Combinatorics, Computing and Complexity, ed. D. Du and G. Hu, Kluwer, 1989. LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 Michael Albert, Michael Engen, Jay Pantone, Vincent Vatter, Universal layered permutations, arXiv:1710.04240 [math.CO], (2017). Michael Albert, Michael Engen, Jay Pantone, Vincent Vatter, Universal Layered Permutations, Electronic Journal of Combinatorics. Volume 25(3), 2018, #P3.23. J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197. Sung-Hyuk Cha, On Integer Sequences Derived from Balanced k-ary Trees, Applied Mathematics in Electrical and Computer Engineering, 2012. Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From N. J. A. Sloane, Dec 24 2012 Hsien-Kuei Hwang, S. Janson, T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016. Hsien-Kuei Hwang, S. Janson, T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585. Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014. D. Knuth, Letter to N. J. A. Sloane, date unknown R. Stephan, Some divide-and-conquer sequences ... R. Stephan, Table of generating functions Eric Weisstein's World of Mathematics, Sorting. FORMULA Let n = 2^(k-1) + g, 0 <= g <= 2^(k-1); then a(n) = 1 + n*k - 2^k. - N. J. A. Sloane, 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, Apr 13 2002 a(1)=0, for n>1, a(n) = ceiling(n*a(n-1)/(n-1)+1). - Benoit Cloitre, Apr 26 2003 a(n) = n-1 + min { a(k)+a(n-k) : 1 <= k <= n-1 }, cf. A003314. - Vladeta Jovovic, Aug 15 2004 a(n) = A061168(n-1) + n - 1 for n>1. - Hieronymus Fischer, Dec 05 2006 a(n) = A123753(n-1) - n. - Peter Luschny, Nov 30 2017 MAPLE a := proc(n) local k; k := ilog2(n) + 1; 1 + n*k - 2^k end; # N. J. A. Sloane, Dec 01 2007 [edited by Peter Luschny, Nov 30 2017] MATHEMATICA a[n_?EvenQ] := a[n] = n + 2a[n/2] - 1; a[n_?OddQ] := a[n] = n + a[(n+1)/2] + a[(n-1)/2] - 1; a = 0; a = 1; Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Nov 23 2011, after Pari *) a[n_] := n IntegerLength[n, 2] - 2^IntegerLength[n, 2] + 1; Table[a[n], {n, 1, 58}] (* Peter Luschny, Dec 02 2017 *) PROG (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) (Haskell) import Data.List (transpose) a001855 n = a001855_list !! n a001855_list = 0 : zipWith (+) [1..] (zipWith (+) hs \$ tail hs) where    hs = concat \$ transpose [a001855_list, a001855_list] -- Reinhard Zumkeller, Jun 03 2013 (Python) def A001855(n):     s, i, z = 0, n-1, 1     while 0 <= i: s += i; i -= z; z += z     return s print([A001855(n) for n in range(1, 59)]) # Peter Luschny, Nov 30 2017 CROSSREFS Partial sums of A029837. Cf. A003071, A000337, A030190, A030308, A061168, A123753. Sequence in context: A130262 A094228 A278586 * A006591 A310027 A310028 Adjacent sequences:  A001852 A001853 A001854 * A001856 A001857 A001858 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Additional comments from M. D. McIlroy (mcilroy(AT)dartmouth.edu) STATUS approved

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Last modified June 17 18:39 EDT 2021. Contains 345085 sequences. (Running on oeis4.)