A006282 Sorting numbers: number of comparisons in Batcher's parallel sort. (Formerly M2447)

0, 1, 3, 5, 9, 12, 16, 19, 26, 31, 37, 41, 48, 53, 59, 63, 74, 82, 91, 97, 107, 114, 122, 127, 138, 146, 155, 161, 171, 178, 186, 191, 207, 219, 232, 241, 255, 265, 276, 283, 298, 309, 321, 329, 342, 351, 361, 367, 383, 395, 408, 417, 431, 441, 452, 459, 474

Offset

1,3

References

R. W. Floyd and D. E. Knuth, The Bose-Nelson sorting problem, pp. 163-172 of J. N. Srivastava, ed., A Survey of Combinatorial Theory, North-Holland, 1973.

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.3.4, Eq. (10).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, preprint.

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

Index entries for sequences related to sorting

Formula

a(1)=0, a(n) = a(ceiling(n/2)) + a(floor(n/2)) + C(ceiling(n/2), floor(n/2)), n > 1, where the C function is defined in Knuth by C(m,n) = m*n if m*n <= 1 and C(m,n) = C(ceiling(m/2),ceiling(n/2)) + C(floor(m/2),floor(n/2)) + floor((m+n-1)/2)) otherwise.

Example

G.f. = x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 12*x^6 + 16*x^7 + 19*x^8 + 26*x^9 + 31*x^10 + ...

Mathematica

c[m_, n_] /; m*n <= 1 = m*n; c[m_, n_] := c[m, n] = c[ Ceiling[m/2], Ceiling[n/2] ] + c[ Floor[m/2], Floor[n/2] ] + Floor[(m + n - 1)/2]; a[1] = 0; a[n_] := a[n] = a[ Ceiling[n/2] ] + a[ Floor[n/2] ] + c[ Ceiling[n/2], Floor[n/2] ]; Table[ a[n], {n, 1, 57}] (* Jean-François Alcover, Jan 19 2012, from formula *)

Prog

(PARI) (c(m, n) = local(i, j); i=ceil(m/2); j=ceil(n/2); if( m*n<2, m*n, c(i, j) + c(m\2, n\2) + (m+n-1)\2)); {a(n) = local(i, j); i=ceil(n/2); j=floor(n/2); if( n<2, 0, a(i) + a(j) + c(i, j))}; /* Michael Somos, Feb 07 2004 */

Crossrefs

Cf. A003075.

First differences are in A083742.

Sequence in context: A287105 A003075 A061562 * A086845 A375649 A243205

Adjacent sequences: A006279 A006280 A006281 * A006283 A006284 A006285

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved