A113098
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 = 2 and t_i = 2 (mod 3) and t_{i+1} <= 4*t_i for 1
Original entry on oeis.org
1, 2, 13, 242, 13228, 2241527, 1237069018, 2305369985312, 14874520949557933, 338242806223319079422, 27474512329417917714396073, 8057337874806992183898478061882, 8607002252619465665736907583406214288
Offset: 0
The tree of 4-tournament sequences of descendents
of a node labeled (2) begins:
[2]; generation 1: 2->[5,8]; generation 2:
5->[8,11,14,17,20], 8->[11,14,17,20,23,26,29,32]; ...
Then a(n) gives the number of nodes in generation n.
Also, a(n+1) = sum of labels of nodes in generation n.
Cf.
A008934,
A113077,
A113078,
A113079,
A113085,
A113089,
A113096,
A113100,
A113107,
A113109,
A113111,
A113113.
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{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^2)[n+1,1])}
A113095
Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^4](n-1,k-1) + [T^4](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^4 is the matrix 4th power of T.
Original entry on oeis.org
1, 1, 1, 4, 5, 1, 46, 66, 21, 1, 1504, 2398, 978, 85, 1, 146821, 255113, 122914, 14962, 341, 1, 45236404, 84425001, 46001193, 7046354, 235122, 1365, 1, 46002427696, 91159696960, 54661544301, 9933169553, 432627794, 3738738, 5461, 1
Offset: 0
Triangle T begins:
1;
1,1;
4,5,1;
46,66,21,1;
1504,2398,978,85,1;
146821,255113,122914,14962,341,1;
45236404,84425001,46001193,7046354,235122,1365,1; ...
Matrix third power T^3 (A113099) begins:
1;
3,1;
27,15,1;
693,513,63,1;
52812,47619,8289,255,1; ...
where column 0 equals A113100.
Matrix 4th power T^4 (A113101) begins:
1;
4,1;
46,20,1;
1504,894,84,1;
146821,108292,14622,340,1;
45236404,39188597,6812596,233758,1364,1; ...
where adjacent sums in row n of T^4 forms row n+1 of T.
-
{T(n,k)=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,k+1])}
Showing 1-2 of 2 results.
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