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Search: id:A123535
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| A123535 |
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Recurrence from values at floor of a third and two-thirds. |
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+0
1
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| 4, 8, 16, 17, 26, 32, 33, 43, 58, 59, 61, 73, 74, 90, 101, 102, 105, 124, 125, 127, 145, 146, 158, 170, 171, 175, 210, 211, 213, 217, 218, 237, 241, 242, 255, 280, 281, 283, 289, 290, 326, 344, 345, 348, 364, 365, 367, 388, 389, 394, 399, 400, 414, 459, 460
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Roughly analogous to maximal number of comparisons for sorting n elements by binary insertion (A001855).
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LINKS
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Wikipedia, Akra-Bazzi method. As of 10 Nov 2006, this article correctly gives the asymptotic, but incorrectly refers to merge-sort rather than binary insertion sort.
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FORMULA
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a(0) = 1, for n>1: a(floor(n/3)) + a(floor(2n/3)) + n + 1.
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EXAMPLE
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a(0) = 1 by definition.
a(1) = a(floor(1/3)) + a(floor(2/3)) + 1 + 1 = a(0) + a(0) + 2 = 4.
a(2) = a(floor(2/3)) + a(floor(4/3)) + 2 + 1 = a(0) + a(1) + 3 = 8.
a(3) = a(floor(3/3)) + a(floor(6/3)) + 3 + 1 = a(1) + a(2) + 4 = 16.
a(4) = a(floor(4/3)) + a(floor(8/3)) + 4 + 1 = a(1) + a(2) + 5 = 17.
a(5) = a(floor(5/3)) + a(floor(10/3)) + 5 + 1 = a(1) + a(3) + 6 = 26.
a(6) = a(floor(6/3)) + a(floor(12/3)) + 6 + 1 = a(2) + a(4) + 7 = 32.
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MAPLE
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A123535 := proc(n) options remember ; if n = 0 then RETURN(1) ; else RETURN(A123535(floor(n/3))+A123535(floor(2*n/3))+n+1) ; fi ; end: for n from 1 to 100 do printf("%d, ", A123535(n)) ; od : - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 13 2007
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CROSSREFS
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Cf. A000040, A001768, A001855, A029837, A003071, A000337, A030190, A030308, A061168.
Sequence in context: A045534 A166634 A072603 this_sequence A065192 A161994 A070738
Adjacent sequences: A123532 A123533 A123534 this_sequence A123536 A123537 A123538
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 11 2006
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EXTENSIONS
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Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 13 2007
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