A113088
Triangle T, read by rows, equal to the matrix square of triangle A113084, which satisfies the recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k).
Original entry on oeis.org
1, 2, 1, 10, 8, 1, 114, 118, 26, 1, 2970, 3668, 1108, 80, 1, 182402, 257122, 96416, 9964, 242, 1, 27392682, 42821472, 18871894, 2501468, 89182, 728, 1, 10390564242, 17650889358, 8826033518, 1412198686, 65914154, 799714, 2186, 1
Offset: 0
Triangle begins:
1;
2,1;
10,8,1;
114,118,26,1;
2970,3668,1108,80,1;
182402,257122,96416,9964,242,1;
27392682,42821472,18871894,2501468,89182,728,1; ...
-
{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^3)[r-1,c-1])+(M^3)[r-1,c]))); return((M^2)[n+1,k+1])}
A113090
Triangle T, read by rows, equal to the matrix cube of triangle A113084, which satisfies the recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k).
Original entry on oeis.org
1, 3, 1, 21, 12, 1, 331, 255, 39, 1, 11973, 11326, 2442, 120, 1, 1030091, 1136709, 310864, 22206, 363, 1, 218626341, 272246616, 89081163, 8266954, 199839, 1092, 1, 118038692523, 162043308555, 61099562421, 6923071251, 220482175, 1796349
Offset: 0
Triangle begins:
1;
3,1;
21,12,1;
331,255,39,1;
11973,11326,2442,120,1;
1030091,1136709,310864,22206,363,1;
218626341,272246616,89081163,8266954,199839,1092,1; ...
-
{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^3)[r-1,c-1])+(M^3)[r-1,c]))); return((M^3)[n+1,k+1])}
A113087
Row sums of triangle A113084, which satisfies the recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k).
Original entry on oeis.org
1, 2, 8, 68, 1252, 51724, 5000468, 1176844012, 696653833108, 1063944196415276, 4273738134967763028, 45828521528368595280876, 1327368163105850592309345044, 104832927462400739554960944906668
Offset: 0
-
{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(sum(k=1,n+1,M[n+1,k]))}
A113082
Main diagonal of square table A113081; also, a(n) equals the n-th term in column 0 of the matrix n-th power of triangle A113084.
Original entry on oeis.org
1, 1, 10, 331, 33476, 10204145, 9378590446, 26026690264407, 218132378185337416, 5518274388618175447069, 421034872020570533423509010, 96809747319527667989371938562883
Offset: 0
-
{a(n,q=3)=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^n)[n+1,1])}
A113083
Diagonal of square table A113081; also, a(n) equals the n-th term in column 0 of the matrix (n+1)-th power of triangle A113084.
Original entry on oeis.org
1, 2, 21, 724, 75695, 23694838, 22239639177, 62747494950248, 532868670719193651, 13624738004791751175370, 1048678107774203901392276461, 242892250870416811233766661498812
Offset: 1
-
{a(n,q=3)=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(if(n<1,0,(M^n)[n,1]))}
A113086
Column 1 of triangle A113084, which satisfies the recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k).
Original entry on oeis.org
1, 4, 33, 586, 23299, 2166800, 490872957, 280082001078, 412989850899863, 1604627387895595612, 16673986511766782602033, 468802076619022792385223170, 35999655498002937223238590342603
Offset: 1
-
{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(if(n<1,0,M[n+1,2]))}
A113089
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 = 2 and t_i = 2 (mod 2) and t_{i+1} <= 3*t_i for 1
Original entry on oeis.org
1, 2, 10, 114, 2970, 182402, 27392682, 10390564242, 10210795262650, 26494519967902114, 184142934938620227530, 3466516611360924222460082, 178346559667060145108789818842, 25264074391478558474014952210052802
Offset: 0
The tree of 3-tournament sequences of even integer
descendents of a node labeled (2) begins:
[2]; generation 1: 2->[4,6];
generation 2: 4->[6,8,10,12], 6->[8,10,12,14,16,18]; ...
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,
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^2)[n+1,1])}
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.
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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^5)[r-1,c-1])+(M^5)[r-1,c]))); return(M[n+1,k+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])}
A113081
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 3-tournament sequences, for n>=1.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 21, 10, 3, 1, 0, 331, 114, 21, 4, 1, 0, 11973, 2970, 331, 36, 5, 1, 0, 1030091, 182402, 11973, 724, 55, 6, 1, 0, 218626341, 27392682, 1030091, 33476, 1345, 78, 7, 1, 0, 118038692523, 10390564242, 218626341, 3697844, 75695, 2246
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,...
0,3,10,21,36,55,78,105,136,171,210,...
0,21,114,331,724,1345,2246,3479,5096,7149,...
0,331,2970,11973,33476,75695,148926,265545,440008,...
0,11973,182402,1030091,3697844,10204145,23694838,...
0,1030091,27392682,218626341,1011973796,3416461455,...
0,218626341,10390564242,118038692523,706848765844,...
0,118038692523,10210795262650,166013096151621,...
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/* Generalized Cook-Kleber Recurrence */ T(n,k,q=3)=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))))))
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/* Matrix Power Recurrence (Paul D. Hanna) */ T(n,k,q=3)=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])
Showing 1-10 of 11 results.
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