A001768 Sorting numbers: number of comparisons for merge insertion sort of n elements. (Formerly M2408 N0954)

0, 1, 3, 5, 7, 10, 13, 16, 19, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 71, 76, 81, 86, 91, 96, 101, 106, 111, 116, 121, 126, 131, 136, 141, 146, 151, 156, 161, 166, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255

Offset

1,3

References

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.3.1.

T. A. J. Nicholson, A method for optimizing permutation problems and its industrial applications, pp. 201-217 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

L. R. Ford, Jr. and S. M. Johnson, A tournament problem, Amer. Math. Monthly, 66 (1959), 387-389.

Eric Weisstein's World of Mathematics, Sorting

Index entries for sequences related to sorting

Formula

a(n) = Sum_{k=1..n} ceiling(log_2 (3k/4)). See also Problem 5.3.1-14 of Knuth.

a(n) = n(z-1)-[(2^(z+2)-3z)/6] where z = [log_2(3n+3)]. - David W. Wilson, Feb 26 2006

Maple

Digits := 60: A001768 := proc(n) local k; add( ceil( log(3*k/4)/log(2) ), k=1..n); end;
# second Maple program:
b:= proc(n) option remember; ceil(log[2](3*n/4)) end:
a:= proc(n) option remember; `if`(n<1, 0, a(n-1)+b(n)) end:
seq(a(n), n=1..61);  # Alois P. Heinz, Dec 03 2019

Mathematica

Accumulate[Ceiling[Log[2, (3*Range[60])/4]]] (* Harvey P. Dale, Oct 30 2013 *)

Prog

(PARI) a(n)=ceil(log(3/4*n)/log(2)) \\ Charles R Greathouse IV, Nov 04 2011
(Haskell)
a001768 n = n * (z - 1) - (2 ^ (z + 2) - 3 * z) `div` 6
   where z = a085423 $ n + 1
-- Reinhard Zumkeller, Mar 16 2013  after David W. Wilson's formula.

Crossrefs

Cf. A085423, A000523.

Sequence in context: A016040 A003070 A036604 * A327672 A329815 A089108

Adjacent sequences: A001765 A001766 A001767 * A001769 A001770 A001771

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Name clarified by Li-yao Xia, Nov 18 2015

Status

approved