A121924 Number of splitting steps that one can take with a sequence of n 2's.

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

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) 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

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