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A121924
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Number of splitting steps that one can take with a sequence of n 2's.
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3
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0, 1, 1, 3, 4, 4, 7, 9, 10, 10, 14, 17, 19, 20, 20, 25, 29, 32, 34, 35, 35, 41, 46, 50, 53, 55, 56, 56, 63, 69, 74, 78, 81, 83, 84, 84, 92, 99, 105, 110, 114, 117, 119, 120, 120, 129, 137, 144, 150, 155, 159, 162, 164, 165, 165, 175, 184, 192, 199, 205, 210, 214, 217
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OFFSET
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1,4
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COMMENTS
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See "A class of trees and its Wiener index" (or Table 2.1 on page 12 of Wagner's PhD thesis) for details. Many of the papers of Stephan Wagner are available at his home page in PDF format.
A splitting step is replacing a pair (c, c) with a pair (c+1, c-1). - Peter Kagey, Sep 24 2017
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LINKS
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FORMULA
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a(n) = binomial(b(n),3) + (n-binomial(b(n),2))*(b(n)^2+3b(n)-2(n+1))/4, where b(n) = floor(sqrt(2n+1/4)+1/2) - Stephan Wagner (swagner(AT)sun.ac.za), Jul 18 2007
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EXAMPLE
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a(11) = 14 from the formula, since b(11) = 5.
For n = 8 an example of a(8) = 9 splitting steps is:
[2 2 2 2 2 2 2 2]
[3 2 2 2 2 2 2 1]
[3 3 2 2 2 2 1 1]
[3 3 3 2 2 1 1 1]
[3 3 3 3 1 1 1 1]
[4 3 3 2 1 1 1 1]
[4 4 2 2 1 1 1 1]
[4 4 3 1 1 1 1 1]
[5 3 3 1 1 1 1 1]
[5 4 2 1 1 1 1 1] (End)
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PROG
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(Haskell)
a121924 n = a007318 b 3 + (n - a007318 b 2) * (b*(b+3) - 2*(n+1)) `div` 4
where b = round $ sqrt $ 2 * fromIntegral n + 1/4
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Edited by Stephan Wagner (swagner(AT)sun.ac.za), Jul 18 2007
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STATUS
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approved
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