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%I #20 Sep 25 2017 06:59:35
%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,
%T 55,56,56,63,69,74,78,81,83,84,84,92,99,105,110,114,117,119,120,120,
%U 129,137,144,150,155,159,162,164,165,165,175,184,192,199,205,210,214,217
%N Number of splitting steps that one can take with a sequence of n 2's.
%C 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.
%C A splitting step is replacing a pair (c, c) with a pair (c+1, c-1). - _Peter Kagey_, Sep 24 2017
%H Reinhard Zumkeller, <a href="/A121924/b121924.txt">Table of n, a(n) for n = 1..10000</a>
%H Robert F. Tichy and Stephan Wagner, <a href="http://www.liebertonline.com/doi/abs/10.1089/cmb.2005.12.1004?journalCode=cmb">Extremal Problems for Topological Indices in Combinatorial Chemistry</a>.
%H Stephan Wagner, <a href="http://finanz.math.tu-graz.ac.at/~wagner/">Home page of Stephan G. Wagner</a>.
%H Stephan Wagner, <a href="http://finanz.math.tu-graz.ac.at/~wagner/pub.html">Publications of Stephan G. Wagner</a>
%H Stephan Wagner, <a href="http://dx.doi.org/10.1007/s10440-006-9026-5">A class of trees and its Wiener index</a>, Acta Applic. Mathem. 91 (2) (2006) 119-132.
%H S. Wagner, <a href="http://finanz.math.tu-graz.ac.at/~wagner/Diss.pdf">Graph-theoretical enumeration and digital expansions: an analytic approach</a>, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. Univ. Graz, Austria, Feb., 2006.
%H S. Wagner and R. F. Tichy, <a href="http://math.sun.ac.za/~swagner/extrtop.pdf">Extremal problems for topological indices in combinatorial chemistry</a>, J. of Computational Biology, vol. 12 (2005), pp. 1004-1013.
%F 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
%e a(11) = 14 from the formula, since b(11) = 5.
%e From _Peter Kagey_, Sep 24 2017 (Start)
%e For n = 8 an example of a(8) = 9 splitting steps is:
%e [2 2 2 2 2 2 2 2]
%e [3 2 2 2 2 2 2 1]
%e [3 3 2 2 2 2 1 1]
%e [3 3 3 2 2 1 1 1]
%e [3 3 3 3 1 1 1 1]
%e [4 3 3 2 1 1 1 1]
%e [4 4 2 2 1 1 1 1]
%e [4 4 3 1 1 1 1 1]
%e [5 3 3 1 1 1 1 1]
%e [5 4 2 1 1 1 1 1] (End)
%o (Haskell)
%o a121924 n = a007318 b 3 + (n - a007318 b 2) * (b*(b+3) - 2*(n+1)) `div` 4
%o where b = round $ sqrt $ 2 * fromIntegral n + 1/4
%o -- _Reinhard Zumkeller_, Sep 02 2013
%Y Cf. A007318.
%K nonn
%O 1,4
%A _Parthasarathy Nambi_, Sep 02 2006
%E Edited by Stephan Wagner (swagner(AT)sun.ac.za), Jul 18 2007
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