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A113089 Number of 3-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 2 and t_i = 2 (mod 2) and t_{i+1} <= 3*t_i for 1

%I A113089 #9 Mar 30 2012 18:36:51
%S A113089 1,2,10,114,2970,182402,27392682,10390564242,10210795262650,
%T A113089 26494519967902114,184142934938620227530,3466516611360924222460082,
%U A113089 178346559667060145108789818842,25264074391478558474014952210052802
%N A113089 Number of 3-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 2 and t_i = 2 (mod 2) and t_{i+1} <= 3*t_i for 1<i<n.
%C A113089 Column 0 of triangle A113088; A113088 is the matrix square of triangle A113084, which satisfies the matrix recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k). Also equals column 2 of square table A113081.
%H A113089 T. D. Noe, <a href="/A113089/b113089.txt">Table of n, a(n) for n=0..30</a>
%H A113089 M. Cook and M. Kleber, <a href="http://www.combinatorics.org/Volume_7/Abstracts/v7i1r44.html">Tournament sequences and Meeussen sequences</a>, Electronic J. Comb. 7 (2000), #R44.
%e A113089 The tree of 3-tournament sequences of even integer
%e A113089 descendents of a node labeled (2) begins:
%e A113089 [2]; generation 1: 2->[4,6];
%e A113089 generation 2: 4->[6,8,10,12], 6->[8,10,12,14,16,18]; ...
%e A113089 Then a(n) gives the number of nodes in generation n.
%e A113089 Also, a(n+1) = sum of labels of nodes in generation n.
%o A113089 (PARI) {a(n)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^3)[r-1,c-1])+(M^3)[r-1,c]))); return((M^2)[n+1,1])}
%Y A113089 Cf. A008934, A113077, A113078, A113079, A113085, A113096, A113098, A113100, A113107, A113109, A113111, A113113.
%K A113089 nonn
%O A113089 0,2
%A A113089 _Paul D. Hanna_, Oct 14 2005