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A113100 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 3 and t_i = 3 (mod 3) and t_{i+1} <= 4*t_i for 1

%I A113100 #9 Mar 30 2012 18:36:51
%S A113100 1,3,27,693,52812,12628008,9924266772,26507035453923,
%T A113100 246323730279500082,8100479557816637139288,
%U A113100 954983717308947379891713642,407790020849346203244152231395953
%N A113100 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 3 and t_i = 3 (mod 3) and t_{i+1} <= 4*t_i for 1<i<n.
%C A113100 Column 0 of triangle A113099; A113099 is the matrix cube of triangle A113095, which satisfies the matrix recurrence: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](n-1,k). Also equals column 3 of square table A113092.
%H A113100 T. D. Noe, <a href="/A113100/b113100.txt">Table of n, a(n) for n=0..30</a>
%H A113100 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 A113100 The tree of 4-tournament sequences of descendents of a node labeled (3) begins:
%e A113100 [3]; generation 1: 3->[6,9,12]; generation 2:
%e A113100 6->[9,12,15,18,21,24], 9->[12,15,18,21,24,27,30,33,36],
%e A113100 12->[15,18,21,24,27,30,33,36,39,42,45,48]; ...
%e A113100 Then a(n) gives the number of nodes in generation n.
%e A113100 Also, a(n+1) = sum of labels of nodes in generation n.
%o A113100 (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^4)[r-1,c-1])+(M^4)[r-1,c]))); return((M^3)[n+1,1])}
%Y A113100 Cf. A008934, A113077, A113078, A113079, A113085, A113089, A113096, A113098, A113107, A113109, A113111, A113113.
%K A113100 nonn
%O A113100 0,2
%A A113100 _Paul D. Hanna_, Oct 14 2005