A376276 Table T(n, k) n > 0, k > 2 read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. The length of the row number n in column k is equal to the n-th k-gonal number A086270.
1, 3, 1, 4, 4, 1, 2, 3, 4, 1, 8, 5, 5, 5, 1, 7, 2, 3, 4, 5, 1, 9, 10, 6, 6, 6, 6, 1, 6, 11, 2, 3, 4, 5, 6, 1, 10, 9, 13, 7, 7, 7, 7, 7, 1, 5, 12, 12, 2, 3, 4, 5, 6, 7, 1, 16, 8, 14, 15, 8, 8, 8, 8, 8, 8, 1, 15, 13, 11, 16, 2, 3, 4, 5, 6, 7, 8, 1, 17, 7, 15, 14, 18, 9, 9, 9, 9, 9, 9, 9, 1, 14, 14, 10, 17, 17, 2, 3, 4, 5, 6, 7, 8, 9, 1, 18, 6, 16, 13, 19, 20, 10, 10, 10
Offset: 1
Examples
Table begins:
k = 3 4 5 6 7 8
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n = 1: 1, 1, 1, 1, 1, 1, ...
n = 2: 3, 4, 4, 5, 5, 6, ...
n = 3: 4, 3, 5, 4, 6, 5, ...
n = 4: 2, 5, 3, 6, 4, 7, ...
n = 5: 8, 2, 6, 3, 7, 4, ...
n = 6: 7, 10, 2, 7, 3, 8, ...
n = 7: 9, 11, 13, 2, 8, 3, ...
n = 8: 6, 9, 12, 15, 2, 9, ...
n = 9: 10, 12, 14, 16, 18, 2, ...
n =10: 5, 8, 11, 14, 17, 20, ...
n =11: 16, 13, 15, 17, 19, 21, ...
n =12: 15, 7, 10, 13, 16, 19, ...
n =13: 17, 14, 16, 18, 20, 22, ...
n =14: 14, 6, 9, 12, 15, 18, ...
n =15: 18, 23, 17, 19, 21, 23, ...
n =16: 13, 22, 8, 11, 14, 17, ...
n =17: 19, 24, 18, 20, 22, 24, ...
n =18: 12, 21, 7, 10, 13, 16, ...
n =19: 20, 25, 30, 21, 23, 25, ...
n =20: 11, 20, 29, 9, 12, 15, ...
... .
For k = 3 the first 4 blocks have lengths 1,3,6 and 10.
For k = 4 the first 3 blocks have lengths 1,4, and 9.
For k = 5 the first 3 blocks have lengths 1,5, and 12.
Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
1;
3, 1;
4, 4, 1;
2, 3, 4, 1;
8, 5, 5, 5, 1;
7, 2, 3, 4, 5, 1;
References
- E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 45.
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.
- Eric Weisstein's World of Mathematics, Polygonal Number.
- Index entries for sequences that are permutations of the natural numbers.
- Index to sequences related to polygonal numbers
Crossrefs
Programs
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Mathematica
T[n_,k_]:=Module[{L,R,P,Res,result},L=Ceiling[Max[x/.NSolve[x^3*(k-2)+3*x^2-x*(k-5)-6*n==0,x,Reals]]]; R=n-(((L-1)^3)*(k-2)+3*(L-1)^2-(L-1)*(k-5))/6;P=Which[OddQ[R]&&OddQ[k*L*(L-1)/2-L^2+2*L],((k*L*(L-1)/2-L^2+2*L+1-R)+1)/2,OddQ[R]&&EvenQ[k*L*(L-1)/2-L^2+2*L],(R+k*L*(L-1)/2-L^2+2*L+1)/2,EvenQ[R]&&OddQ[k*L*(L-1)/2-L^2+2*L],Ceiling[(k*L*(L-1)/2-L^2+2*L+1)/2]+R/2,EvenQ[R]&&EvenQ[k*L*(L-1)/2-L^2+2*L],Ceiling[(k*L*(L-1)/2-L^2+2*L+1)/2]-R/2]; Res=P+((L-1)^3*(k-2)+3*(L-1)^2-(L-1)*(k-5))/6;result=Res;result] Nmax=6;Table[T[n,k],{n,1,Nmax},{k,3,Nmax+2}]
Formula
T(n,k) = P(n,k) + ((L(n,k)-1)^3*(k-2)+3*(L(n,k)-1)^2-(L(n,k)-1)*(k-5))/6, where L(n,k) = ceiling(x(n,k)), x(n,k) is largest real root of the equation x^3*(k - 2) + 3*x^2 - x*(k - 5) - 6*n = 0. R(n,k) = n - ((L(n,k) - 1)^3*(k-2)+3*(L(n,k)-1)^2-(L(n,k)-1)*(k-5))/6. P(n,k) = ((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 2 - R(n,k)) / 2 if R is odd and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is odd, P(n,k) = (R(n,k) + (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2 if R is odd and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is even, P(n,k) = ceiling(((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2) + (R(n,k) / 2) if R is even and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is odd, P(n,k) = ceiling(((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2) - (R(n,k) / 2) if R is even and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is even.
Comments