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
Original entry on oeis.org
1, 3, 27, 693, 52812, 12628008, 9924266772, 26507035453923, 246323730279500082, 8100479557816637139288, 954983717308947379891713642, 407790020849346203244152231395953
Offset: 0
The tree of 4-tournament sequences of descendents of a node labeled (3) begins:
[3]; generation 1: 3->[6,9,12]; generation 2:
6->[9,12,15,18,21,24], 9->[12,15,18,21,24,27,30,33,36],
12->[15,18,21,24,27,30,33,36,39,42,45,48]; ...
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,
A113098,
A113107,
A113109,
A113111,
A113113.
-
{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])}
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])}
A113097
Triangle T, read by rows, equal to the matrix square of triangle A113095, which satisfies the recurrence: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](n-1,k).
Original entry on oeis.org
1, 2, 1, 13, 10, 1, 242, 237, 42, 1, 13228, 15296, 3741, 170, 1, 2241527, 2930006, 893528, 58909, 682, 1, 1237069018, 1775967132, 637702746, 54501208, 935709, 2730, 1, 2305369985312, 3563503353790, 1451785389252, 151058838746
Offset: 0
Triangle begins:
1;
2,1;
13,10,1;
242,237,42,1;
13228,15296,3741,170,1;
2241527,2930006,893528,58909,682,1;
1237069018,1775967132,637702746,54501208,935709,2730,1; ...
-
{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^2)[n+1,k+1])}
Showing 1-3 of 3 results.
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