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A003071 Sorting numbers: maximal number of comparisons for sorting n elements by list merging.
(Formerly M2443)
20
0, 1, 3, 5, 9, 11, 14, 17, 25, 27, 30, 33, 38, 41, 45, 49, 65, 67, 70, 73, 78, 81, 85, 89, 98, 101, 105, 109, 115, 119, 124, 129, 161, 163, 166, 169, 174, 177, 181, 185, 194, 197, 201, 205, 211, 215, 220, 225, 242, 245, 249, 253, 259, 263, 268, 273, 283, 287, 292, 297, 304 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Comment from Jeremy Gardiner, Dec 28 2008: The following sequences all appear to have the same parity: A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975.

a(A092246(n)) = A230720(n); a(A230709(n)) = A230721(n+1). - Reinhard Zumkeller, Oct 28 2013

REFERENCES

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

D. E. Knuth, Art of Computer Programming, Vol. 3, Sections 5.2.4 and 5.3.1.

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

LINKS

Moshe Levin, Table of n, a(n) for n = 1..10000

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

Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193, 2014

Index entries for sequences related to sorting

FORMULA

Let n = 2^e_1 + 2^e_2 + ... + 2^e_t, e_1 > e_2 > ... > e_t >= 0, t >= 1. Then a(n) = 1 - 2^e_t + Sum_{1<=k<=t} (e_k+k-1)*2^e_k [Knuth, Problem 14, Section 5.2.4].

a(n) = a(n-1) + A061338(n) = a(n-1) + A006519(n) + A000120(n)-1 = n + A000337(A000523(n)) + a(n-2^A000523(n)). a(2^k) = k*2^k + 1 = A002064(k). - Henry Bottomley, Apr 27 2001

G.f.: x/(1-x)^3 + 1/(1-x)^2*Sum(k>=1, (-1+(1-x)*2^(k-1))*x^2^k/(1-x^2^k)). - Ralf Stephan, Apr 17 2003

MATHEMATICA

a[1] = 0; a[n_] := a[n] = a[n-1] + 2^IntegerExponent[n-1, 2] + DigitCount[n-1, 2, 1] - 1; Table[a[n], {n, 1, 61}] (* Jean-Fran├žois Alcover, Feb 10 2012, after Henry Bottomley *)

PROG

(Haskell)

a003071 n = 1 - 2 ^ last es +

   sum (zipWith (*) (zipWith (+) es [0..]) (map (2 ^) es))

   where es = reverse $ a133457_row n

-- Reinhard Zumkeller, Oct 28 2013

CROSSREFS

Cf. A001855.

Cf. A133457.

Sequence in context: A139099 A152259 A219611 * A178442 A284310 A109324

Adjacent sequences:  A003068 A003069 A003070 * A003072 A003073 A003074

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified August 18 14:32 EDT 2017. Contains 290720 sequences.