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A121924
Number of splitting steps that one can take with a sequence of n 2's.
3
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
OFFSET
1,4
COMMENTS
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
LINKS
Stephan Wagner, A class of trees and its Wiener index, Acta Applic. Mathem. 91 (2) (2006) 119-132.
S. Wagner, Graph-theoretical enumeration and digital expansions: an analytic approach, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. Univ. Graz, Austria, Feb., 2006.
S. Wagner and R. F. Tichy, Extremal problems for topological indices in combinatorial chemistry, J. of Computational Biology, vol. 12 (2005), pp. 1004-1013.
FORMULA
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
EXAMPLE
a(11) = 14 from the formula, since b(11) = 5.
From Peter Kagey, Sep 24 2017 (Start)
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)
PROG
(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
-- Reinhard Zumkeller, Sep 02 2013
CROSSREFS
Cf. A007318.
Sequence in context: A154426 A231219 A231343 * A241740 A342332 A225738
KEYWORD
nonn
AUTHOR
Parthasarathy Nambi, Sep 02 2006
EXTENSIONS
Edited by Stephan Wagner (swagner(AT)sun.ac.za), Jul 18 2007
STATUS
approved