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URL: https://oeis.org/A003071

⇱ A003071 - OEIS

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A003071
Sorting numbers: maximal number of comparisons for sorting n elements by list merging.
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
OFFSET
1,3
COMMENTS
The following sequences all appear to have the same parity: A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975. - Jeremy Gardiner, Dec 28 2008
a(A092246(n)) = A230720(n); a(A230709(n)) = A230721(n+1). - Reinhard Zumkeller, Oct 28 2013
REFERENCES
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
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.
An Vinh Nguyen Dinh, Nhien Pham Hoang Bao, Terrillon Jean-Christophe, Hiroyuki Iida, Reaper Tournament System, 2018.
An Vinh Nguyen Dinh, Nhien Pham Hoang Bao, Mohd Nor Akmal Khalid, Hiroyuki Iida, Simulating competitiveness and precision in a tournament structure: a reaper tournament system, Int'l J. of Information Technology (2020) Vol. 12, 1-18.
Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.
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_{k=1..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-François 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
Sequence in context: A139099 A152259 A219611 * A178442 A319986 A359408
KEYWORD
nonn,easy,nice
EXTENSIONS
More terms from David W. Wilson
STATUS
approved