A113106
Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^5](n-1,k-1) + [T^5](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^5 is the matrix 5th power of T.
Original entry on oeis.org
1, 1, 1, 5, 6, 1, 85, 115, 31, 1, 4985, 7420, 2590, 156, 1, 1082905, 1744965, 723370, 62090, 781, 1, 930005021, 1601759426, 752616215, 82390620, 1532715, 3906, 1, 3306859233805, 6024941167511, 3117415999361, 409321203715, 10025307495
Offset: 0
Triangle begins:
1;
1,1;
5,6,1;
85,115,31,1;
4985,7420,2590,156,1;
1082905,1744965,723370,62090,781,1;
930005021,1601759426,752616215,82390620,1532715,3906,1;
Matrix 4th power T^4 (A113112) begins:
1;
4,1;
56,24,1;
2704,1576,124,1;
481376,346624,39376,624,1; ...
where column 0 equals A113113.
Matrix 5th power T^5 (A113114) begins:
1;
5,1;
85,30,1;
4985,2435,155,1;
1082905,662060,61310,780,1;
930005021,671754405,80861810,1528810,3905,1; ...
where adjacent sums in row n of T^5 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^5)[r-1,c-1])+(M^5)[r-1,c]))); return(M[n+1,k+1])}
A113111
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
Original entry on oeis.org
1, 3, 33, 1251, 173505, 94216515, 210576669921, 2002383115518243, 82856383278525698433, 15166287556997012904054915, 12437232461209961704387810340769
Offset: 0
The tree of 5-tournament sequences of descendents
of a node labeled (3) begins:
[3]; generation 1: 3->[7,11,15];
generation 2: 7->[11,15,19,23,27,31,35],
11->[15,19,23,27,31,35,39,43,47,51,55],
15->[19,23,27,31,35,39,43,47,51,55,59,63,67,71,75]; ...
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,
A113100,
A113107,
A113109,
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^5)[r-1,c-1])+(M^5)[r-1,c]))); return((M^3)[n+1,1])}
A113112
Triangle T, read by rows, equal to the matrix 4th power of triangle A113106, which satisfies the recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k).
Original entry on oeis.org
1, 4, 1, 56, 24, 1, 2704, 1576, 124, 1, 481376, 346624, 39376, 624, 1, 337587520, 284081376, 41686624, 979376, 3124, 1, 978162377600, 927672109184, 165184873376, 5122890624, 24434376, 15624, 1, 12088945462984960
Offset: 0
Triangle begins:
1;
4,1;
56,24,1;
2704,1576,124,1;
481376,346624,39376,624,1;
337587520,284081376,41686624,979376,3124,1;
978162377600,927672109184,165184873376,5122890624,24434376,15624,1;
-
{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^5)[r-1,c-1])+(M^5)[r-1,c]))); return((M^4)[n+1,k+1])}
A113114
Triangle T, read by rows, equal to the matrix 5th power of triangle A113106, which satisfies the recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k).
Original entry on oeis.org
1, 5, 1, 85, 30, 1, 4985, 2435, 155, 1, 1082905, 662060, 61310, 780, 1, 930005021, 671754405, 80861810, 1528810, 3905, 1, 3306859233805, 2718081933706, 399334065655, 9987138060, 38169435, 19530, 1, 50220281721033905
Offset: 0
Triangle begins:
1;
5,1;
85,30,1;
4985,2435,155,1;
1082905,662060,61310,780,1;
930005021,671754405,80861810,1528810,3905,1;
3306859233805,2718081933706,399334065655,9987138060,38169435,19530,1;
-
{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^5)[r-1,c-1])+(M^5)[r-1,c]))); return((M^5)[n+1,k+1])}
Showing 1-4 of 4 results.
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