

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
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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, c1).  Peter Kagey, Sep 24 2017


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert F. Tichy and Stephan Wagner, Extremal Problems for Topological Indices in Combinatorial Chemistry.
Stephan Wagner, Home page of Stephan G. Wagner.
Stephan Wagner, Publications of Stephan G. Wagner
Stephan Wagner, A class of trees and its Wiener index, Acta Applic. Mathem. 91 (2) (2006) 119132.
S. Wagner, Graphtheoretical 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. 10041013.


FORMULA

a(n) = binomial(b(n),3) + (nbinomial(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
Adjacent sequences: A121921 A121922 A121923 * A121925 A121926 A121927


KEYWORD

nonn


AUTHOR

Parthasarathy Nambi, Sep 02 2006


EXTENSIONS

Edited by Stephan Wagner (swagner(AT)sun.ac.za), Jul 18 2007


STATUS

approved



