A113096 - OEIS

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

%S 1,1,4,46,1504,146821,45236404,46002427696,159443238441379,

%T 1926751765436372746,82540801108546193896804,

%U 12696517688186899788062326096,7084402815778394692932546017050054

%N 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 = 1 and t_i = 1 (mod 3) and t_{i+1} <= 4*t_i for 1<i<n.

%C Equals column 0 of triangle A113095, which satisfies: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](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 4-tournament sequences of descendents

%e of a node labeled (1) begins:

%e [1]; generation 1: 1->[4]; generation 2: 4->[7,10,13,16];

%e generation 3: 7->[10,13,16,19,22,25,28],

%e 10->[13,16,19,22,25,28,31,34,37,40],

%e 13->[16,19,22,25,28,31,34,37,40,43,46,49,52],

%e 16->[19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64]; ...

%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^4)[r-1,c-1])+(M^4)[r-1,c]))); return(M[n+1,1])}

%Y Cf. A008934, A113077, A113078, A113079, A113085, A113089, A113098, A113100, A113107, A113109, A113111, A113113.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 14 2005