Boris Putievskiy - cp's OEIS frontend

A383722 a(n) = A378762(A382679(n)).

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

Offset: 1

Boris Putievskiy, May 07 2025

This sequence can be regarded as 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 also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.

			Triangle array begins:
  k=    1   2   3   4   5  6   7   8   9
  n=1:  1;
  n=2:  5,  3,  6,  2,  4;
  n=3:  14, 8, 12, 10, 15, 9, 13, 7, 11;
(1, 5, 3, ..., 7, 11) = (1, 2, 3, ..., 12, 11) (1, 5, 3, ..., 7, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A382679.
(1, 5, 3, ..., 7, 11) = (1, 5, 3, ..., 7, 11)^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4, 10, 11, ...
   5,  2, 12,  7, 23, ...
   6,  8, 13, 19, 24, ...
  14,  9, 25, 18, 40, ...
  15, 17, 26, 32, 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;
  4, 2, 5, 1, 3;
  8, 2, 6, 4, 9, 3, 7, 1, 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), A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

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

A383723 a(n) = A378762(A376214(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, May 07 2025

This sequence can be regarded as 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 also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.

			Triangle array begins:
  k=    1  2  3   4   5   6   7   8   9
  n=1:  1;
  n=2:  2, 3, 6,  5,  4;
  n=3:  9, 8, 7, 10, 15, 12, 13, 14, 11;
(1, 2, 3, ..., 14, 11) = (1, 2, 3, ..., 12, 11) (1, 2, 3, ..., 12, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A376214.
(1, 2, 3, ..., 14, 11) = (1, 2, 3, ..., 14, 11)^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4, 10, 11, ...
   2,  5,  7, 14, 16, ...
   6,  8, 13, 19, 24, ...
   9, 12, 18, 25, 31, ...
  15, 17, 26, 32, 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;
  1, 2, 5, 4, 3;
  3, 2, 1, 4, 9, 6, 7, 8, 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), A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

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

A383724 a(n) = A378762(A382680(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, May 07 2025

This sequence can be regarded as 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 also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.

			Triangle array begins:
  k=     1  2   3   4   5  6   7  8   9
  n=1:   1;
  n=2:   5, 3,  6,  2,  4;
  n=3:  12, 8, 14, 10, 15, 7, 13, 9, 11;
(1, 5, 3, ..., 9, 11) = (1, 2, 3, ..., 12, 11) (1, 5, 3, ..., 9, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A382680.
(1, 5, 3, ..., 9, 11) = (1, 5, 3, ..., 9, 11)^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4, 10, 11, ...
   5,  2, 14,  9, 27, ...
   6,  8, 13, 19, 24, ...
  12,  7, 25, 18, 42, ...
  15, 17, 26, 32, 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;
  4, 2, 5, 1, 3;
  6, 2, 8, 4, 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), A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

T(n,k) for 1 <= k <= 4n - 3: 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) = k if k < m and k == 0 (mod 2), P(n, k) = 3m - 1 - k 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.

A383590 a(n) = A378762(A382499(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, May 01 2025

This sequence can be regarded as 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 also be regarded 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 the Cantor numbering method.
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A383419 (the inverse permutation), A381968, A381662, A382499, A380817, A382679, A376214, A382680, A378762, A383589, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy link. - Boris Putievskiy, Jun 02 2025

			Triangle array begins:
  k=    1   2   3  4   5  6   7  8   9
  n=1:  1;
  n=2:  5,  3,  6, 2,  4;
  n=3: 14, 10, 12, 8, 15, 7, 13, 9, 11;
(1, 5, 3, ..., 9, 11) = (1, 2, 3, ..., 12, 11) (1, 5, 3, ..., 7, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A382499.
Ord(1, 5, 3, ... , 9, 11) = 4.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4,  8, 11, ...
   5,  2, 12,  9, 23, ...
   6, 10, 13, 19, 24, ...
  14,  7, 25, 18, 40, ...
  15, 21, 26, 34, 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;
  4, 2, 5, 1, 3;
  8, 4, 6, 2, 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.
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.

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

A383589 a(n) = A378762(A381662(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, May 01 2025

This sequence can be regarded as 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 also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.

			Triangle array begins:
  k=    1   2  3  4   5   6   7   8   9
  n=1:  1;
  n=2:  2,  3, 6, 5,  4;
  n=3:  7, 10, 9, 8, 15, 12, 13, 14, 11;
(1, 2, 3, ..., 14, 11) = (1, 2, 3, ..., 12, 11) (1, 2, 3, ..., 12, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A381662.
(1, 2, 3, ..., 14, 11) = (1, 2, 3, ..., 14, 11)^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4,  8, 11, ...
   2,  5,  9, 14, 20, ...
   6, 10, 13, 19, 24, ...
   7, 12, 18, 25, 33, ...
  15, 21, 26, 34, 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;
  1, 2, 5, 4, 3;
  1, 4, 3, 2, 9, 6, 7, 8, 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), A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

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

A383419 a(n) = A378762(A381968(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, May 01 2025

This sequence can be regarded as 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 also be regarded 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 the Cantor numbering method.
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A383590 (the inverse permutation), A381968, A381662, A382499, A380817, A382679, A376214, A382680, A378762, A383589, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy link. - Boris Putievskiy, Jun 02 2025

			Triangle array begins:
  k=    1   2   3  4   5  6   7  8   9
  n=1:  1;
  n=2:  5,  3,  6, 2,  4;
  n=3: 12, 10, 14, 8, 15, 9, 13, 7, 11;
(1, 5, 3, ..., 7, 11) = (1, 2, 3, ..., 12, 11) (1, 5, 3, ..., 9, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A381968.
Ord(1, 5, 3, ..., 7, 11) = 4.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4,  8, 11, ...
   5,  2, 14,  7, 27, ...
   6, 10, 13, 19, 24, ...
  12,  9, 25, 18, 42, ...
  15, 21, 26, 34, 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;
  4, 2, 5, 1, 3;
  6, 4, 8, 2, 9, 3, 7, 1, 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.
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.

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

A382680 a(n) = A382499(A380817(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Apr 03 2025

This sequence can be regarded as 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 also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A381968 and  and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral Group D4. The list of the 8 elements of that group: this sequence, A382679 (the inverse permutation), A000027 (the identity permutation), A381968, A381662, A382499, A380817, A376214. For subgroups and the Cayley table of the group D4 see Putievskiy (D4 (I)) link. - Boris Putievskiy, Apr 27 2025
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A380817, A382679, A376214, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Putievskiy link. - Boris Putievskiy, Jun 02 2025

			Triangle array begins:
  k=    1   2   3  4   5  6   7  8   9
  n=1:  1;
  n=2:  5,  3,  4, 2,  6;
  n=3: 12, 10, 14, 8, 11, 7, 13, 9, 15;
(1,5,3,...,9,15) = (1,5,3,...,9,15)^(-1).
(1,5,3,...,9,15) = (1,5,3,...,7,15) (1,2,3,...,10,7,...,14,15). The first permutation on the right-hand side is from Example A382499 and the second from Example A380817.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
    1,  3,  6,  8, 15, ...
    5,  2, 14,  9, 27, ...
    4, 10, 13, 19, 26, ...
   12,  7, 25, 18, 42, ...
   11, 21, 24, 34, 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,
  4, 2, 3, 1, 5,
  6, 4, 8, 2, 5, 1, 7, 3, 9.

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.
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table.
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

T(n,k) for 1 <= k <= 4n - 3: 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) = k 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.

A382679 a(n) = A381968(A380817(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Apr 03 2025

This sequence can be regarded as 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 also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A381968 and  and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral Group D4. The list of the 8 elements of that group: this sequence, A382680 (the inverse permutation), A000027 (the identity permutation), A381968, A381662, A382499, A380817, A376214. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - Boris Putievskiy, Apr 27 2025
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A380817, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, 383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy (2025 D4xC2) link. - Boris Putievskiy, May 27 2025

			Triangle array begins:
  k=    1   2   3  4   5  6   7  8   9
  n=1:  1;
  n=2:  5,  3,  4, 2,  6;
  n=3: 14, 10, 12, 8, 11, 9, 13, 7, 15;
(1,5,3,...,7,15) = (1,5,3,...,7,15)^(-1).
(1,5,3,...,7,15) = (1,5,3,...,9,15) (1,2,3,...,10,7,...,14,15).  The first permutation on the right-hand side is from Example A381968 and the second from Example A380817.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6,  8, 15, ...
   5,  2, 12,  7, 23, ...
   4, 10, 13, 19, 26, ...
  14,  9, 25, 18, 40, ...
  11, 21, 24, 34, 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;
  4, 2, 3, 1, 5;
  8, 4, 6, 2, 5, 3, 7, 1, 9.

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.
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (2025 D4 (I)).
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table (2025 D4xC2).
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

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

where m = 2n - 1.

A382499 Inverse permutation to A381968.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Mar 29 2025

This sequence can be regarded as 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 also be regarded 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 the Cantor numbering method.
A381968 and  and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A381968 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - Boris Putievskiy, Apr 17 2025
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968 (the inverse permutation), A381662, A380817, A382679, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy (2025 D4xC2) link. - Boris Putievskiy, May 27 2025

			Triangle array begins:
  k=     1  2   3   4   5  6   7  8   9
  n=1:   1;
  n=2:   5, 3,  4,  2,  6;
  n=3:  12, 8, 14, 10, 11, 9, 13, 7, 15;
(1,5,3,...,7,15)^(-1) = (1,5,3,...,9,15). The permutation on the right-hand side is from Example A381968.
ord(1,5,3,...,7,15) = 4.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6, 10, 15, ...
   5,  2, 14,  7, 27, ...
   4,  8, 13, 19, 26, ...
  12,  9, 25, 18, 42, ...
  11, 17, 24, 32, 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;
  4, 2, 3, 1, 5;
  6, 2, 8, 4, 5, 3, 7, 1, 9.

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.
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (2025 D4 (I)).
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table (2025 D4xC2).
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.

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

A381968 a(a(n)) = A381662(n).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Mar 12 2025

This sequence can be regarded as 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 also be regarded 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 the Cantor numbering method.
This sequence and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A382499 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Putievskiy (D4 (I)) link. - Boris Putievskiy, Jun 09 2025
This sequence,  A378762 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A382499 (the inverse permutation), A381662, A380817, A382679, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Putievskiy link. - Boris Putievskiy, Jun 09 2025

			Triangle array begins:
  k=     1  2   3   4   5   6   7   8   9
  n=1:   1;
  n=2:   5, 3,  4,  2,  6;
  n=3:  14, 8, 12, 10, 11,  7, 13,  9, 15;
(1,5,3,...,9,15) (1,5,3,...,9,15) = (1,2,3,...,12,15). The permutation on the right-hand side is from Example A381662.
ord(1,5,3,...,9,15) = 4.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6, 10, 15, ...
   5,  2, 12,  9, 23, ...
   4,  8, 13, 19, 26, ...
  14,  7, 25, 18, 40, ...
  11, 17, 24, 32, 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;
  4, 2, 3, 1, 5;
  8, 2, 6, 4, 5, 1, 7, 3, 9.

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.
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (D4 (I)).
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.

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

cp's OEIS Frontend

User: Boris Putievskiy

A383722 a(n) = A378762(A382679(n)).

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A383723 a(n) = A378762(A376214(n)).

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A383724 a(n) = A378762(A382680(n)).

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A383590 a(n) = A378762(A382499(n)).

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A383589 a(n) = A378762(A381662(n)).

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A383419 a(n) = A378762(A381968(n)).

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A382680 a(n) = A382499(A380817(n)).

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A382679 a(n) = A381968(A380817(n)).