A113109
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 = 2 and t_i = 2 (mod 4) and t_{i+1} <= 5*t_i for 1
Original entry on oeis.org
1, 2, 16, 440, 43600, 16698560, 26098464448, 172513149018752, 4938593053649344000, 622793203804403960906240, 350552003258337075784341271552, 890153650520295355798989668668129280
Offset: 0
The tree of 5-tournament sequences of descendents
of a node labeled (2) begins:
[2]; generation 1: 2->[6,10]; generation 2:
6->[10,14,18,22,26,30], 10->[14,18,22,26,30,34,38,42,46,50]; ...
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,
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^5)[r-1,c-1])+(M^5)[r-1,c]))); return((M^2)[n+1,1])}
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.
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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^5)[r-1,c-1])+(M^5)[r-1,c]))); return((M^3)[n+1,1])}
A113103
Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of 5-tournament sequences.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 5, 2, 1, 0, 85, 16, 3, 1, 0, 4985, 440, 33, 4, 1, 0, 1082905, 43600, 1251, 56, 5, 1, 0, 930005021, 16698560, 173505, 2704, 85, 6, 1, 0, 3306859233805, 26098464448, 94216515, 481376, 4985, 120, 7, 1, 0, 50220281721033905
Offset: 0
Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,...
0,1,2,3,4,5,6,7,8,9,10,11,...
0,5,16,33,56,85,120,161,208,261,320,...
0,85,440,1251,2704,4985,8280,12775,18656,26109,...
0,4985,43600,173505,481376,1082905,2122800,3774785,6241600,...
0,1082905,16698560,94216515,337587520,930005021,2156566656,...
0,930005021,26098464448,210576669921,978162377600,...
0,3306859233805,172513149018752,2002383115518243,...
0,50220281721033905,4938593053649344000,82856383278525698433,...
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/* Generalized Cook-Kleber Recurrence */
{T(n,k,q=5)=if(n==0,1,if(n<0||k<=0,0,if(n==1,k, if(n>=k,sum(j=1,k,T(n-1,k+(q-1)*j)), sum(j=1,n+1,(-1)^(j-1)*binomial(n+1,j)*T(n,k-j))))))}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
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/* Matrix Power Recurrence (Paul D. Hanna) */
{T(n,k,q=5)=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^q)[r-1,c-1])+(M^q)[r-1,c]))); (M^k)[n+1,1]}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
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