A370655 -id:A370655 - cp's OEIS frontend

A373498 a(a(a(n))) = A370655(n).

2, 1, 3, 5, 9, 7, 6, 8, 4, 10, 12, 18, 14, 20, 16, 15, 17, 13, 19, 11, 21, 23, 31, 25, 33, 27, 35, 29, 28, 30, 26, 32, 24, 34, 22, 36, 38, 48, 40, 50, 42, 52, 44, 54, 46, 45, 47, 43, 49, 41, 51, 39, 53, 37, 55

Offset: 1

Boris Putievskiy, Jun 17 2024

Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is an intra-block permutation of positive integers.
The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 =  A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6.  - Boris Putievskiy, Aug 02 2024

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   2,  1,  3;
  n=2:   5,  9,  7,  6,  8,  4, 10;
  n=3:  12, 18, 14, 20, 16, 15, 17, 13, 19, 11, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
   2,  3,  7, 10, 16, 21, ...
   1,  9,  4, 20, 11, 35, ...
   5,  8, 14, 19, 27, 34, ...
   6, 18, 13, 33, 24, 52, ...
  12, 17, 25, 32, 42, 51, ...
  15, 31, 26, 50, 41, 73, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
   2,1,3,
   2,6,4,3,5,1,7,
   2,8,4,10,6,5,7,3,9,1,11,
   ...
The 3rd power of each permutation is equal to the corresponding permutation in example A370655:
   (2,1,3)^3 = (2,1,3),
   (2,6,4,3,5,1,7)^3 = (1,2,4,3,5,6,7),
   (2,8,4,10,6,5,7,3,9,1,11)^3 = (3,4,1,2,6,5,7,10,9,8,11).

Boris Putievskiy, Table of n, a(n) for n = 1..9870 _Boris Putievskiy_, Aug 02 2024
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
_Boris Putievskiy_, Aug 02 2024

Cf. A000027, A004767 (row lengths), A204164, A370655, A374447, A374494, A374531.

Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1);
P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}]
Table[a[a[a[n]]],{n,1,Nmax}] (* A370655 *)

Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n), R(n) = n - (L(n)-1)*(2*L(n)-1),
P(n) = R(n) + 1 if R(n) <= 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 2*L(n) + R(n) if R(n) <= 2*L(n)-1 and R(n) mod 2 = 0, P(n) = R(n) if R(n) > 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 4*L(n) - R(n) - 1 if R(n) > 2*L(n)-1 and R(n) mod 2 = 0.
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = k + 1 if k <= 2*n-1 and k mod 2 = 1, P(n,k) = 2*n + k if k <= 2*n-1 and k mod 2 = 0,
P(n,k) = k if k > 2*n-1 and k mod 2 = 1, P(n,k) = 4*n - k - 1 if k > 2*n-1 and k mod 2 = 0.

A204164 Symmetric matrix based on f(i,j) = floor((i+j)/2), by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Clark Kimberling, Jan 12 2012

A204164 represents the matrix M given by f(i,j) = floor((i+j)/2) for i >= 1 and j >= 1.  See A204165 for characteristic polynomials of principal submatrices of M, with interlacing zeros.  See A204016 for a guide to other choices of M.
k appears 4k-1 times, k > 0. - Boris Putievskiy, Jun 12 2024
Number of numbers of the form 2k^2+k+1 <= n, for k = 0,1,2,... - Wesley Ivan Hurt, Jun 19 2024

			Northwest corner:
  1 1 2 2 3 3 4 4
  1 2 2 3 3 4 4 5
  2 2 3 3 4 4 5 5
  2 3 3 4 4 5 5 6
  3 3 4 4 5 5 6 6

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.

Cf. A057211, A204165, A204016, A202453, A370655.

f[i_, j_] := Floor[(i + j)/2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i], {n, 1, 15}, {i, 1, n}]]  (* this sequence *)
  (* or *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                 (* A204165 *)
TableForm[Table[c[n], {n, 1, 10}]]
  (* or *)
a[n_] = Ceiling[(Sqrt[8*n + 1] - 1)/4];
Nmax = 21; Table[a[n], {n, 1, Nmax}] (* Boris Putievskiy, Jun 12 2024 *)

from math import isqrt
def A204164(n): return (m:=isqrt(n>>1))+(n>m*((m<<1)+1)) # Chai Wah Wu, Nov 14 2024

a(n) = ceiling((sqrt(8*n+1)-1)/4). - Boris Putievskiy, Jun 12 2024
a(n) = Sum_{k=1..n} [c(k) = c(k-1)+1], where c(n) = floor(sqrt(2n)+1/2) mod 2 = A057211(n) and [] is the Iverson bracket. - Wesley Ivan Hurt, Jun 23 2024
a(n) = m+1 if n>m(2m+1) and a(n) = m otherwise where m = floor(sqrt(n/2)). - Chai Wah Wu, Nov 14 2024

A374447 Inverse permutation to A373498.

Original entry on oeis.org

2, 1, 3, 9, 4, 7, 6, 8, 5, 10, 20, 11, 18, 13, 16, 15, 17, 12, 19, 14, 21, 35, 22, 33, 24, 31, 26, 29, 28, 30, 23, 32, 25, 34, 27, 36, 54, 37, 52, 39, 50, 41, 48, 43, 46, 45, 47, 38, 49, 40, 51, 42, 53, 44, 55

Offset: 1

Boris Putievskiy, Jul 08 2024

Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is an intra-block permutation of positive integers.
The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6. - Boris Putievskiy, Aug 03 2024

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   2,  1,  3;
  n=2:   9,  4,  7,  6,  8,  5, 10;
  n=3:  20, 11, 18, 13, 16, 15, 17, 12, 19, 14, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
    2,  3,  7, 10, 16, 21, ...
    1,  4,  5, 13, 14, 26, ...
    9,  8, 18, 19, 31, 34, ...
    6, 11, 12, 24, 25, 41, ...
   20, 17, 33, 32, 50, 51, ...
   15, 22, 23, 39, 40. 60, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
    2,1,3,
    6,1,4,3,5,2,7,
   10,1,8,3,6,5,7,2,9,4,11
   ...
The inverse permutation of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^(-1) = (2,1,3),
   (2,6,4,3,5,1,7)^(-1) = (6,1,4,3,5,2,7),
   (2,8,4,10,6,5,7,3,9,1,11)^(-1) = (10,1,8,3,6,5,7,2,9,4,11).
The 5th power of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^5 = (2,1,3),
   (2,6,4,3,5,1,7)^5 = (6,1,4,3,5,2,7),
   (2,8,4,10,6,5,7,3,9,1,11)^5 = (10,1,8,3,6,5,7,2,9,4,11).

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

Cf. A000027, A004767 (row lengths), A204164, A370655, A373498, A374494, A374531.

Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1); P=Which[R<=2*L&&Mod[R,2]==1,4*L-R-1,R<=2*L&&Mod[R,2]==0,R-1,R>2*L&&Mod[R,2]==1,R,R>2*L&&Mod[R,2]==0,-2*L+R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}]
Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1); P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}] (* A373498 *)
Table[a[a[a[a[a[n]]]]],{n,1,Nmax}] (* this sequence *)

Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n), P(n) = 4*L(n) - R(n) - 1, if R(n) <= 2*L(n) and R(n) mod 2 = 1, P(n) = R(n) - 1, if R(n) <= 2*L(n) and R(n) mod 2 = 0, P(n) = R(n), if R(n) > 2*L(n) and R(n) mod 2 = 1, P(n) = - 2*L(n) + R(n), if R(n) > 2*L(n) and R(n) mod 2 = 0.
a(n) = A373498(A373498(A373498(A373498(A373498(n))))).
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = 4*n - k - 1, if k <= 2*n and k mod 2 = 1, P(n,k) = k-1, if k <= 2*n and k mod 2 = 0, P(n,k) = k, if k >  2*n and k mod 2 = 1, P(n,k) = -2*n + k, if k >  2*n and k mod 2 = 0.

A374494 a(n) = A373498(A373498(n)).

Original entry on oeis.org

1, 2, 3, 9, 4, 6, 7, 8, 5, 10, 18, 13, 20, 11, 15, 16, 17, 14, 19, 12, 21, 31, 26, 33, 24, 35, 22, 28, 29, 30, 27, 32, 25, 34, 23, 36, 48, 43, 50, 41, 52, 39, 54, 37, 45, 46, 47, 44, 49, 42, 51, 40, 53, 38, 55

Offset: 1

Boris Putievskiy, Jul 09 2024

Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is an intra-block permutation of positive integers.
The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6.  - Boris Putievskiy, Aug 02 2024

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   1,  2,  3;
  n=2:   9,  4,  6,  7,  8,  5, 10;
  n=3:  18, 13, 20, 11, 15, 16, 17, 14, 19, 12, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
   1,  3,  6, 10, 15, 21, ...
   2,  4,  5, 11, 12, 22, ...
   9,  8, 20, 19, 35, 34, ...
   7, 13, 14, 24, 25, 39, ...
  18, 17, 33, 32, 52, 51, ...
  16, 26, 27, 41, 42, 60, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
   1,2,3,
   6,1,3,4,5,2,7,
   8,3,10,1,5,6,7,4,9,2,11
   ...
The 2nd power of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^2 = (1,2,3),
   (2,6,4,3,5,1,7)^2 = (6,1,3,4,5,2,7),
   (2,8,4,10,6,5,7,3,9,1,11)^2 = (8,3,10,1,5,6,7,4,9,2,11).

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.

Cf. A000027, A004767 (row lengths), A204164, A370655, A373498, A374447, A374531.

Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1);
P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}] (* A373498 *)
Table[a[a[n]],{n,1,Nmax}] (* this sequence *)
Nmax = 21;
a[n_] := Module[{L, R, P, Result}, L = Ceiling[(Sqrt[8*n + 1] - 1)/4];
  R = n - (L - 1)*(2*L - 1);
  P = Which[R < 2*L - 1 && Mod[R, 2] == 1, 2*L + R + 1, R < 2*L - 1 && Mod[R, 2] == 0, 2*L - R - 1, R >= 2*L - 1 && Mod[R, 2] == 1, R, R >= 2*L - 1 && Mod[R, 2] == 0, 4*L - R];
  Result = P + (L - 1)*(2*L - 1);
  Result]
Table[a[n], {n, 1, Nmax}]

Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n), R(n) = n - (L(n)-1)*(2*L(n)-1), P(n) = 2*L(n) + R(n) + 1 if R(n) < 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 2*L(n) - R(n) - 1 if R(n) < 2*L(n)-1 and R(n) mod 2 = 0, P(n) = R(n) if R(n) >= 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 4*L(n) - R(n) if R(n) >= 2*L(n)-1 and R(n) mod 2 = 0.
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = 2*n + k + 1 if k < 2*n-1 and k mod 2 = 1, P(n,k) = 2*n - k - 1 if k < 2*n-1 and k mod 2 = 0, P(n,k) = k if k >= 2*n-1 and k mod 2 = 1, P(n,k) = 4*n - k if k >= 2*n-1 and k mod 2 = 0.

A374531 a(n) = A374494(A374494(n)).

Original entry on oeis.org

1, 2, 3, 5, 9, 6, 7, 8, 4, 10, 14, 20, 12, 18, 15, 16, 17, 11, 19, 13, 21, 27, 35, 25, 33, 23, 31, 28, 29, 30, 22, 32, 24, 34, 26, 36, 44, 54, 42, 52, 40, 50, 38, 48, 45, 46, 47, 37, 49, 39, 51, 41, 53, 43, 55

Offset: 1

Boris Putievskiy, Jul 10 2024

Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is an intra-block permutation of positive integers.
The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6.  - Boris Putievskiy, Aug 02 2024

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   1,  2,  3;
  n=2:   5,  9,  6,  7,  8,  4, 10;
  n=3:  14, 20, 12, 18, 15, 16, 17, 11, 19, 13, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
   1,  3,  6, 10, 15, 21, ...
   2,  9,  4, 18, 13, 31, ...
   5,  8, 12, 19, 23, 34, ...
   7, 20, 11, 33, 24, 50, ...
  14, 17, 25, 32, 40, 51, ...
  16, 35, 22, 52, 39, 73, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
   1,2,3,
   2,6,3,4,5,1,7
   4,10,2,8,5,6,7,1,9,3,11
   ...
The 4th power of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^4 = (1,2,3),
   (2,6,4,3,5,1,7)^4 = (2,6,3,4,5,1,7),
   (2,8,4,10,6,5,7,3,9,1,11)^4 = (4,10,2,8,5,6,7,1,9,3,11).

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.

Cf. A000027, A004767 (row lengths), A204164, A370655, A373498, A374447, A374494.

Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1);
P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}] (* A373498(n) *)
Table[a[a[a[a[n]]]],{n,1,Nmax}] (* this sequence *)
Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1);
P=Which[R<2*L-1&&Mod[R,2]==1,2*L-R-1,R<2*L-1&&Mod[R,2]==0,4*L-R,R==2*L,R,R>=2*L-1&&Mod[R,2]==1,R,R>=2*L-1&&Mod[R,2]==0,-2*L+R-1];
Result=P+(L-1)*(2*L-1);
Result];
Table[a[n],{n,1,Nmax}]

Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n), R(n) = n - (L(n)-1)*(2*L(n)-1), P(n) = 2*L(n) - R(n) - 1, if R(n) < 2*L(n) - 1 and R(n) mod 2 = 1, P(n) = 4*L(n) - R(n), if R(n) < 2*L(n) - 1 and R(n) mod 2 = 0, P(n) = R(n), if R(n) = 2*L(n), P(n) = R(n), if R(n) >= 2*L(n) - 1 and R(n) mod 2 = 1, P(n) = - 2*L(n) + R(n) - 1, if R(n) >= 2*L(n) - 1 and R(n) mod 2 = 0.
a(n) = A373498(A373498(A373498(A373498(n)))). Boris Putievskiy, Aug 02 2024
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = 2n - k - 1, if k < 2*n-1 and k mod 2 = 1, P(n,k) = 4n - k, if k < 2*n-1 and k mod 2 = 0, P(n,k) = k, if k = 2*n, P(n,k) = k, if k >= 2*n   and k mod 2 = 1, P(n,k) = - 2n + k - 1,   if k >= 2*n   and k mod 2 = 0.

A378200 Square array read by upward antidiagonals: T(n,k) = ((k + n - 1)^2 + (k - n + 1)(-1)^n + (1 - k - n)(-1)^k + (1 - k - n)*(-1)^(k + n) + 2)/2.

Original entry on oeis.org

1, 5, 2, 6, 3, 4, 12, 9, 14, 7, 15, 8, 13, 10, 11, 23, 20, 25, 18, 27, 16, 28, 17, 26, 19, 24, 21, 22, 38, 35, 40, 33, 42, 31, 44, 29, 45, 30, 43, 32, 41, 34, 39, 36, 37, 57, 54, 59, 52, 61, 50, 63, 48, 65, 46, 66, 47, 64, 49, 62, 51, 60, 53, 58, 55, 56

Offset: 1

Boris Putievskiy, Nov 19 2024

The sequence can be arranged in a triangular array, read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n has a length of 4n-3 = A016813(n-1), n > 0. For n > 1, each row is a pair consecutive antidiagonals.
The sequence is an intra-block permutation of the positive integers.
Generalization of Cantor numbering method.
This sequence generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of this sequence with itself: A378684(n) = a(a(n)) = a^2(n), A378762(n) = a^3(n), A379342(n) = a^4(n), A378705(n) = a^5(n). The identity element is A000027(n) = a^6(n). - Boris Putievskiy, Jan 03 2025
A379343 and A378684 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the alternating group A4. The list of the 12 elements of that group: this sequence, A379343 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A378684, A379342, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Mar 28 2025

			Table begins:
   1,  2,  4,  7, 11, ...
   5,  3, 14, 10, 27, ...
   6,  9, 13, 18, 24, ...
  12,  8, 25, 19, 42, ...
  15, 20, 26, 33, 41, ...
  ...
The first 5 antidiagonals are:
   1;
   5, 2;
   6, 3,  4;
  12, 9, 14,  7;
  15, 8, 13, 10, 11;
Triangle array begins:
  k=   1   2   3   4   5   6   7   8   9
  n=1: 1;
  n=2: 5,  2,  6,  3,  4;
  n=3: 12, 9, 14,  7,  15, 8, 13, 10, 11;
Subtracting (n-1)*(2*n-3) from each term is row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  4, 1, 5, 2, 3;
  6, 3, 8, 1, 9, 2, 7, 4, 5.

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.
Boris Putievskiy, The Alternating Group A4: Subgroups and the Cayley Table (2025).
Eric Weisstein's World of Mathematics, Alternating Group.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A370655, A373498, A374447, A374494, A374531, A375602, A375725, A376214, A378684, A378705, A378762, A379342, A379343, A380245, A380815, A380817, A381662, A381663, A381664..

T[n_, k_] := ((k + n - 1)^2 + (k - n + 1)*(-1)^n + (1 - k - n)*(-1)^k + (1 - k - n)*(-1)^(k + n) + 2)/2;
Table[T[k, n], {k, 1, 5}, {n, 1, 5}]

Triangular array T(n,k) for 1 <= k <= 4n - 3 (see Example): T(n,k) = A000384(n-1) + P(n,k), P(n,k) = m + k if k < m and k == 1 (mod 2), P(n,k) = m - k if k < m and k == 0 (mod 2), P(n,k) = 3m - k - 1 if k >= m and k == 1 (mod 2), P(n,k)= - m + k + 1 if k >= m and k == 0 (mod 2), where m = 2n - 1.

G.f.: x*y*(1 + y + y^3 + y^4 - x^3*(2 + y + y^2 - 5*y^3 + y^4) + x*(4 - 3*y + y^2 - y^3 - y^4) - x^2*(1 - 3*y + 9*y^2 + 3*y^3 - 2*y^4) + x^4*(2 + y^2 - 2*y^3 + 3*y^4))/((1 - x)^3*(1 + x)^2*(1 - y)^3*(1 + y)^2). - Stefano Spezia, Jan 12 2025

A378705 Inverse permutation to A378200.

Original entry on oeis.org

1, 3, 5, 6, 2, 4, 10, 12, 8, 14, 15, 7, 13, 9, 11, 21, 23, 19, 25, 17, 27, 28, 16, 26, 18, 24, 20, 22, 36, 38, 34, 40, 32, 42, 30, 44, 45, 29, 43, 31, 41, 33, 39, 35, 37, 55, 57, 53, 59, 51, 61, 49, 63, 47, 65, 66, 46, 64, 48, 62, 50, 60, 52, 58, 54, 56

Offset: 1

Boris Putievskiy, Dec 05 2024

The sequence can be arranged in a triangular array, read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can be represented as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
The sequence is an intra-block permutation of the positive integers.
Generalization of Cantor numbering method.
The sequence A378200 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A378200 with itself: A378684(n) = A378200(A378200(n)) = A378200(n)^2, A378762(n) = A378200(n)^3, A379342(n) = A378200(n)^4, A378705(n) = A378200(n)^5. The identity element is A000027(n) = A378200(n)^6. - Boris Putievskiy, Jan 12 2025

			Triangle array begins:
  k=    1   2   3   4   5   6   7   8   9
  n=1:  1;
  n=2:  3,  5,  6,  2,  4;
  n=3: 10, 12,  8, 14, 15,  7, 13,  9, 11;
(1, 3, 5, ..., 9, 11) = (A378200(1), A378200(2), A378200(3), ..., A378200(14), A378200(15))^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  5,  4, 14, 11, ...
   3,  2,  8,  9, 17, ...
   6, 12, 13, 25, 24, ...
  10,  7, 19, 18, 32, ...
  15, 23, 26, 40, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  2, 4, 5, 1, 3;
  4, 6, 2, 8, 9, 1, 7, 3, 5.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
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.

Cf. A000027, A000384, A016813 (row lengths), A370655, A373498, A374447, A374494, A374531, A375602, A375725, A378200, A378684, A378762, A379342.

Mathematica
```
P[n_,k_]:=Module[{m=2*n-1},If[k
				
```

Linear sequence: (a(1), a(2), ..., a(A000384(n+1))) is permutation of the positive integers from 1 to A000384(n+1). (a(1), a(2), ..., a(A000384(n+1))) = (A378200(1), A378200(2), ..., A378200(A000384(n+1)))^(-1).

Triangular array T(n,k) for 1 <= k <= 4n - 3 (see Example): T(n,k) = A000384(n-1) + P(n,k), P(n,k) = m - k if k < m and k == 1 (mod 2), P(n,k) = m + k - 1 if k < m and k == 0 (mod 2), P(n,k) = 3m - k - 1 if k >= m and k == 1 (mod 2), P(n,k) = - m + k if k >= m and k == 0 (mod 2), where m = 2n - 1.

A378198 Table T(n, k) read by upward antidiagonals. T(n,1) = A375602(n), T(n,2) = A375602(A375602(n)), T(n,3) = A375602(A375602(A375602(n))) and so on.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 3, 3, 2, 1, 5, 4, 4, 2, 1, 6, 5, 3, 3, 2, 1, 7, 6, 5, 4, 4, 2, 1, 10, 7, 6, 5, 3, 3, 2, 1, 13, 16, 7, 6, 5, 4, 4, 2, 1, 16, 14, 9, 7, 6, 5, 3, 3, 2, 1, 8, 9, 17, 13, 7, 6, 5, 4, 4, 2, 1, 11, 10, 13, 12, 14, 7, 6, 5, 3, 3, 2, 1, 14, 8, 16, 14, 11, 17, 7, 6, 5, 4, 4, 2, 1, 17, 17, 10, 9, 17, 8, 12, 7, 6, 5, 3, 3, 2, 1, 19, 12, 12, 16, 13, 12, 10, 11, 7

Offset: 1

Boris Putievskiy, Nov 19 2024

Sequence A375602 generates an infinite cyclic group under composition. The identity element is A000027.
Each column is triangle read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n  has length n(n^2 + 1)/2 = A006003(n).
Each column is an intra-block permutation of the positive integers.
For n > 1, each row combines n consecutive antidiagonals.
Generalization of the Cantor numbering method.

			Table begins:
  k =      1   2   3   4   5   6
--------------------------------------
  n =  1:  1,  1,  1,  1,  1,  1, ...
  n =  2:  2,  2,  2,  2,  2,  2, ...
  n =  3:  4,  3,  4,  3,  4,  3, ...
  n =  4:  3,  4,  3,  4,  3,  4, ...
  n =  5:  5,  5,  5,  5,  5,  5, ...
  n =  6:  6,  6,  6,  6,  6,  6, ...
  n =  7:  7,  7,  7,  7,  7,  7, ...
  n =  8: 10, 16,  9, 13, 14, 17, ...
  n =  9: 13, 14, 17, 12, 11,  8, ...
  n = 10: 16,  9, 13, 14, 17, 12, ...
    ...
Column k = 1 contains the start of A375602.
Ord(T(1,1),T(2,1), ... T(7,1)) = 2, ord(T(1,1),T(2,1), ... T(21,1)) = 18, ord(T(1,1),T(2,1), ... T(55,1)) = 1980, ord(T(1,1),T(2,1), ... T(120,1)) = 51480, where ord is order of permutation.
The first 6 antidiagonals are:
  1;
  2, 1;
  4, 2, 1;
  3, 3, 2, 1;
  5, 4, 4, 2, 1;
  6, 5, 3, 3, 2, 1;

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.

Cf. A000027, A006003 (row lengths), A002817, A370655, A373498, A374447, A374494, A374531, A375602, A375725.

a[n_]:=Module[{L,Ld,Rd,P,Result},L=Ceiling[(Sqrt[4*Sqrt[8*n+1]-3]-1)/2]; Ld=Ceiling[(Sqrt[8*n+1]-1)/2]; Rd=n-(Ld-1)*Ld/2; P=L*Rd+Ld-L*(L+1)/2-Max[Rd-(L^2-L+2)/2,0]*(Max[Rd-(L^2-L+2)/2,0]+1)/2; Result=P+(L-1)*L*(L^2-L+2)/8; Result] (*A375602*) composeSequence[a_,n_,k_]:=Nest[a,n,k]
Nmax=10; Kmax=6; T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Kmax}]

(T(1,k),T(2,k), ... T(A002817(n),k)) is permutation of the integers from 1 to A002817(n). (T(1,k),T(2,k), ... T(A002817(n),k)) = (T(1,1),T(2,1), ... T(A002817(n),1))^k.

cp's OEIS Frontend

A373498 a(a(a(n))) = A370655(n).

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A204164 Symmetric matrix based on f(i,j) = floor((i+j)/2), by antidiagonals.

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A374447 Inverse permutation to A373498.

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A374494 a(n) = A373498(A373498(n)).

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A374531 a(n) = A374494(A374494(n)).

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A378200 Square array read by upward antidiagonals: T(n,k) = ((k + n - 1)^2 + (k - n + 1)*(-1)^n + (1 - k - n)*(-1)^k + (1 - k - n)*(-1)^(k + n) + 2)/2.

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A378705 Inverse permutation to A378200.

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A378198 Table T(n, k) read by upward antidiagonals. T(n,1) = A375602(n), T(n,2) = A375602(A375602(n)), T(n,3) = A375602(A375602(A375602(n))) and so on.

A378200 Square array read by upward antidiagonals: T(n,k) = ((k + n - 1)^2 + (k - n + 1)(-1)^n + (1 - k - n)(-1)^k + (1 - k - n)*(-1)^(k + n) + 2)/2.