A113085
Number of 3-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 2) and t_{i+1} <= 3*t_i for 1
Original entry on oeis.org
1, 1, 3, 21, 331, 11973, 1030091, 218626341, 118038692523, 166013096151621, 619176055256353291, 6207997057962300681573, 169117528577725378851523691, 12626174170113987651028630856581, 2602022118010488151483064379958957003
Offset: 0
The tree of 3-tournament sequences of odd integer
descendents of a node labeled (1) begins:
[1]; generation 1: 1->[3]; generation 2: 3->[5,7,9];
generation 3: 5->[7,9,11,13,15], 7->[9,11,13,15,17,19,21],
9->[11,13,15,17,19,21,23,25,27]; ...
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,
A113089,
A113096,
A113098,
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^3)[r-1,c-1])+(M^3)[r-1,c]))); return(M[n+1,1])}
A113092
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 4-tournament sequences.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 46, 13, 3, 1, 0, 1504, 242, 27, 4, 1, 0, 146821, 13228, 693, 46, 5, 1, 0, 45236404, 2241527, 52812, 1504, 70, 6, 1, 0, 46002427696, 1237069018, 12628008, 146821, 2780, 99, 7, 1, 0, 159443238441379, 2305369985312, 9924266772
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,...
0,4,13,27,46,70,99,133,172,216,265,...
0,46,242,693,1504,2780,4626,7147,10448,14634,...
0,1504,13228,52812,146821,330745,648999,1154923,1910782,...
0,146821,2241527,12628008,45236404,124626530,289031301,...
0,45236404,1237069018,9924266772,46002427696,155367674020,...
0,46002427696,2305369985312,26507035453923,159443238441379,...
0,159443238441379,14874520949557933,246323730279500082,...
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/* Generalized Cook-Kleber Recurrence */
{T(n,k,q=4)=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=4)=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]))); return((M^k)[n+1,1])}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
A113101
Triangle T, read by rows, equal to the matrix 4th power 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, 4, 1, 46, 20, 1, 1504, 894, 84, 1, 146821, 108292, 14622, 340, 1, 45236404, 39188597, 6812596, 233758, 1364, 1, 46002427696, 45157269264, 9504275037, 428894516, 3733278, 5460, 1, 159443238441379, 172969059719500
Offset: 0
Triangle begins:
1;
4,1;
46,20,1;
1504,894,84,1;
146821,108292,14622,340,1;
45236404,39188597,6812596,233758,1364,1;
46002427696,45157269264,9504275037,428894516,3733278,5460,1;
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{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^4)[n+1,k+1])}
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