A113111 - OEIS

%I #5 Mar 30 2012 18:36:51

%S 1,3,33,1251,173505,94216515,210576669921,2002383115518243,

%T 82856383278525698433,15166287556997012904054915,

%U 12437232461209961704387810340769

%N Number of 5-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 4) and t_{i+1} <= 5*t_i for 1<i<n.

%C Equals column 0 of triangle A113110, which is the matrix cube of triangle A113106, which satisfies the recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k).

%H 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 The tree of 5-tournament sequences of descendents

%e of a node labeled (3) begins:

%e [3]; generation 1: 3->[7,11,15];

%e generation 2: 7->[11,15,19,23,27,31,35],

%e 11->[15,19,23,27,31,35,39,43,47,51,55],

%e 15->[19,23,27,31,35,39,43,47,51,55,59,63,67,71,75]; ...

%e Then a(n) gives the number of nodes in generation n.

%e Also, a(n+1) = sum of labels of nodes in generation n.

%o (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^5)[r-1,c-1])+(M^5)[r-1,c]))); return((M^3)[n+1,1])}

%Y Cf. A008934, A113077, A113078, A113079, A113085, A113089, A113096, A113098, A113100, A113107, A113109, A113113.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 14 2005