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A003071
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Sorting numbers: maximal number of comparisons for sorting n elements by list merging.
(Formerly M2443)
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13
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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;
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refs;
listen;
history;
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internal format)
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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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).
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LINKS
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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.
Index entries for sequences related to sorting
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FORMULA
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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
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MATHEMATICA
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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}] (* From Jean-François Alcover, Feb 10 2012, after Henry Bottomley *)
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CROSSREFS
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Cf. A001855.
Sequence in context: A139099 A152259 A219611 * A178442 A109324 A190844
Adjacent sequences: A003068 A003069 A003070 * A003072 A003073 A003074
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from David W. Wilson
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STATUS
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approved
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