A375797 -id:A375797 - cp's OEIS frontend

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

Boris Putievskiy, Sep 18 2024

A209278 presents an algorithm for generating permutations.

The sequence is an intra-block permutation of integer positive numbers.

			Table begins:
  k =      3   4   5   6   7   8
--------------------------------------
  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;

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 45.

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

Cf. A000012, A004526, A028242, A074279, A084964, A086270, A209278, A360010, A375725, A375797

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}]

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.

T(1,n) = A000012(n). T(2,n) = A004526(n+7). T(3,n) = A028242(n+6). T(4,n) = A084964(n+5). T(n-2,n) = A000027(n) for n > 3. L(n,3) = A360010(n). L(n,4) = A074279(n).

A376353 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-pyramidal number A261720.

Original entry on oeis.org

1, 4, 1, 3, 4, 1, 5, 5, 5, 1, 2, 3, 4, 5, 1, 11, 6, 6, 6, 6, 1, 10, 2, 3, 4, 5, 6, 1, 12, 14, 7, 7, 7, 7, 7, 1, 9, 13, 2, 3, 4, 5, 6, 7, 1, 13, 15, 17, 8, 8, 8, 8, 8, 8, 1, 8, 12, 16, 2, 3, 4, 5, 6, 7, 8, 1, 14, 16, 18, 20, 9, 9, 9, 9, 9, 9, 9, 1, 7, 11, 15, 19, 2, 3, 4, 5, 6, 7, 8, 9, 1, 15, 17, 19, 21, 23, 10, 10, 10, 10, 10, 10, 10, 10, 1, 6, 10, 14, 18, 22, 2, 3

Offset: 1

Boris Putievskiy, Sep 21 2024

A209278 presents an algorithm for generating permutations.

The sequence is an intra-block permutation of integer positive numbers.

			Table begins:
  k =      3   4   5   6   7   8
--------------------------------------
  n = 1:   1,  1,  1,  1,  1,  1, ...
  n = 2:   4,  4,  5,  5,  6,  6, ...
  n = 3:   3,  5,  4,  6,  5,  7, ...
  n = 4:   5,  3,  6,  4,  7,  5, ...
  n = 5:   2,  6,  3,  7,  4,  8, ...
  n = 6:  11,  2,  7,  3,  8,  4, ...
  n = 7:  10, 14,  2,  8,  3,  9, ...
  n = 8:  12, 13, 17,  2,  9,  3, ...
  n = 9:   9, 15, 16, 20,  2, 10, ...
  n = 10: 13, 12, 18, 19, 23,  2, ...
  n = 11:  8, 16, 15, 21, 22, 26, ...
  n = 12: 14, 11, 19, 18, 24, 25, ...
  n = 12:  7, 17, 14, 22, 21, 27, ...
  n = 14: 15, 10, 20, 17, 25, 24, ...
  n = 15:  6, 18, 13, 23, 20, 28, ...
          ... .
For k = 3 the first 3 blocks have lengths 1,4 and 10.
For k = 4 the first 2 blocks have lengths 1 and 5.
For k = 5 the first 2 blocks have lengths 1 and 6.
Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
   1;
   4, 1;
   3, 4, 1;
   5, 5, 5, 1;
   2, 3, 4, 5, 1;
  11, 6, 6, 6, 6, 1;

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.

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, Pyramidal Number.
Index entries for sequences that are permutations of the natural numbers.
Index to sequences related to pyramidal numbers.

Cf. A000012, A000027, A004526, A028242, A084964, A168230, A209278, A261720, A375725, A375797.

T[n_,k_]:=Module[{L,R,result},L=Ceiling[Max[x/.NSolve[(k-2)*x^4+2*k*x^3+(14-k)*x^2+(12-2*k)*x-24*n==0,x,Reals]]]; R=n-((k-2)*(L-1)^4+2*k*(L-1)^3+(14-k)*(L-1)^2+(12-2*k)*(L-1))/24; P=Which[OddQ[R]&&OddQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],((L^3*(k-2)+3*L^2-L*(k-5))/6+2-R)/2,OddQ[R]&&EvenQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],(R+(L^3*(k-2)+3*L^2-L*(k-5))/6+1)/2,EvenQ[R]&&OddQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],Ceiling[((L^3*(k-2)+3*L^2-L*(k-5))/6+1)/2]+R/2,EvenQ[R]&&EvenQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],Ceiling[((L^3*(k-2)+3*L^2-L*(k-5))/6+1)/2]-R/2]; Res= P +((k-2)*(L-1)^4+2*k*(L-1)^3+(14-k)*(L-1)^2+(12-2*k)*(L-1))/24; result=Res] Nmax=6; Table[T[n,k],{n,1,Nmax},{k,3,Nmax+2}]

T(n,k) = P(n,k) + ((k-2)*(L(n,k)-1)^4+2*k*(L(n,k)-1)^3+(14-k)*(L(n,k)-1)^2+(12-2*k)*(L(n,k)-1))/24, where L(n,k) = ceiling(x(n,k)), x(n,k) is largest real root of the equation (k-2)*x^4+2*k*x^3+(14-k)*x^2+(12-2*k)*x-24*n = 0. R(n,k) = n - ((k-2)*(L(n,k)-1)^4+2*k*(L(n,k)-1)^3+(14-k)*(L(n,k)-1)^2+(12-2*k)*(L(n,k)-1))/24. P(n,k) = ((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+2-R(n,k))/2 if R(n,k) is odd and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is odd, P(n,k) = ((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+1)+R(n,k))/2 if R(n,k) is odd and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is even, P = ceiling(((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+1)/2)+R(n,k)/2) if R(n,k) is even and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is odd, P = ceiling(((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+1)/2)-R(n,k)/2) if R(n,k) is even and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is even.

T(1,n) = A000012(n). T(2,n) = A004526(n+8). T(3,n) = A028242(n+7). T(4,n) = A084964(n+6). T(5,n) = A168230(n+5). T(n-2,n) = 4*A000012(n) for n > 3. T(n-1,n) = A000027(n) for n > 2.

cp's OEIS Frontend

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.

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