A375725 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows, where each row is a permutation of the numbers of its constituents; see Comments.
1, 3, 1, 2, 2, 6, 4, 3, 2, 10, 5, 4, 4, 2, 1, 6, 9, 3, 8, 14, 1, 10, 6, 5, 4, 3, 20, 28, 8, 7, 1, 6, 12, 3, 2, 36, 9, 8, 7, 5, 5, 18, 26, 2, 1, 7, 5, 20, 7, 10, 5, 4, 34, 44, 1
Offset: 1
Examples
Table begins:
k= 1 2 3 4 5 6
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n= 1: 1, 1, 6, 10, 1, 1, ...
n= 2: 3, 2, 2, 2, 14, 20, ...
n= 3: 2, 3, 4, 8, 3, 3, ...
n= 4: 4, 4, 3, 4, 12, 18, ...
n= 5: 5, 9, 5, 6, 5, 5, ...
n= 6: 6, 6, 1, 5, 10, 16, ...
n= 7: 10, 7, 7, 7, 7, 7, ...
n= 8: 8, 8, 20, 3, 8, 14, ...
n= 9: 9, 5, 9, 9, 9, 9, ...
n=10: 7, 10, 18, 1, 6, 12, ...
n=11: 11, 11, 11, 36, 11, 11, ...
n=12: 14, 20, 16, 12, 4, 10, ...
n=13: 13, 13, 13, 34, 13, 13, ...
n=14: 12, 18, 14, 14, 2, 8, ...
n=15: 15, 15, 15, 32, 15, 15, ...
n=16: 21, 16, 12, 16, 55, 6, ...
n=17: 17, 17, 17, 30, 17, 17, ...
n=18: 19, 14, 10, 18, 53, 4, ...
n=19: 18, 19, 19, 28, 19, 19, ...
n=20: 20, 12, 8, 20, 51, 2, ...
n=21: 16, 21, 21, 26, 21, 21, ...
... .
In column 2, the first 3 blocks have lengths 3,7 and 11. In column 3, the first 2 blocks have lengths 6 and 15. In column 6, the first block has a length of 21.
Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
1;
3, 1;
2, 2, 6;
4, 3, 2, 10;
5, 4, 4, 2, 1;
6, 9, 3, 8, 14, 1;
Links
- Boris Putievskiy, Table of n, a(n) for n = 1..9870
- Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
- Index entries for sequences that are permutations of the natural numbers.
Programs
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Mathematica
T[n_,k_]:=Module[{L,R,P,result},L=Ceiling[(Sqrt[8*n+1]-1)/(2*k)]; R=n-k*(L-1)*(k(L-1)+1)/2; If[2*R>=k^2*L-k*(k-1)/2+1,P=-Max[R,k^2*L-k*(k-1)/2+1-R]*((-1)^R-1)/2+Min[R,k^2*L-k*(k-1)/2+1-R]*((-1)^R+1)/2,P=Max[R,k^2*L-k*(k-1)/2+1-R]*((-1)^(k^2*L-k*(k-1)/2+1-R)+1)/2-Min[R,k^2*L-k*(k-1)/2+1-R]*((-1)^(k^2*L-k*(k-1)/2+1-R)-1)/2]; result=P+k*(L-1)*(k*(L-1)+1)/2] Nmax=21; Table[T[n,k],{n,1,Nmax},{k,1,Nmax}]
Formula
T(n,k) = P(n,k) + k*(L(n,k) - 1)*(k*(L(n,k) - 1) + 1)/2 = P(n,k) + A342719(L(n,k) - 1,k)), where L(n,k) = ceiling((sqrt(8*n+1)-1)/(2*k)), R(n,k) = n - k*(L(n,k)-1)*(k*(L(n,k)-1)+1)/2, P(n,k) = - max(R(n,k) , k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) * ((-1)^R(n,k) - 1) / 2 + min(R(n,k) , k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) * ((-1)^R(n,k) + 1) / 2 if 2R(n,k) ≥k^2*L - k(k-1)/2 + 1, P(n,k) = max(R(n,k) , k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) * ((-1)^(k^2*L(n,k) - k(k-1)/2 + 1 - R) + 1) / 2 - min(R, k^2*L(n,k) - k(k-1)/2 + 1 - R) * ((-1)^(k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) - 1) / 2 if 2R < k^2*L(n,k) - k(k-1)/2. + 1.
Comments