A378705 -id:A378705 - cp's OEIS frontend

A378684 a(n) = A378200(A378200(n)).

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

Offset: 1

Boris Putievskiy, Dec 03 2024

The 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 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 10 2025
This sequence and A379343 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, A379342 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A379343, A380200, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Mar 28 2025

			Triangle array begins:
  k=   1   2   3   4   5   6   7   8   9
  n=1: 1;
  n=2: 3,  5,  4,  2,  6;
  n=3: 8, 14, 10, 12, 11,  9, 13,  7, 15;
(1, 3, 5, ..., 7, 15) = (A378200(1), A378200(2),  A378200(3), ..., A378200(14), A378200(15))^2.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  5,  6, 12, 15, ...
   3,  2, 10,  7, 21, ...
   4, 14, 13, 25, 26, ...
   8,  9, 19, 18, 34, ...
  11, 27, 24, 42, 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, 3, 1, 5;
  2, 8, 4, 6, 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 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), A376214, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664.

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)))^2.

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

A379342 a(n) = A378684(A378684(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Dec 21 2024

The 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 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 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, A378684 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A379343, A380200, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Mar 30 2025

			Triangle array begins:
  k=   1   2   3   4   5   6   7   8   9
  n=1: 1;
  n=2: 5,  2,  4,  3,  6;
  n=3: 14, 7, 12,  9, 11, 10, 13,  8, 15;
  ...
(1, 5, 2, ..., 8, 15) = (A378684(1), A378684(2), A378684(3), ..., A378684(14), A378684(15))^2.
(1, 5, 2, ..., 8, 15) = (A378684(1), A378684(2), A378684(3), ..., A378684(14), A378684(15))^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  2,  6,  9, 15, ...
   5,  3, 12,  8, 23, ...
   4,  7, 13, 18, 26, ...
  14, 10, 25, 19, 40, ...
  11, 16, 24, 31, 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, 1, 3, 2, 5;
  8, 1, 6, 3, 5, 4, 7, 2, 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 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), A376214, A378684, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664.

P[n_,k_]:=If[OddQ[k],Max[k,4 n-3-k],Min[k-1,4 n-2-k]]
Nmax=3;Flatten[Table[P[n,k]+(n-1)*(2*n-3),{n,1,Nmax},{k,1,4*n-3}]]

Linear sequence: (a(1), a(2), ..., a(A000384(n+1))) is a permutation of the positive integers from 1 to A000384(n+1). ord(a(1), a(2), ..., a(A000384(n+1))) = 3, where ord is the order of the permutation. (a(1), a(2), ..., a(A000384(n+1))) = (A378684(1), A378684(2), ..., A378684(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) = max(k, 4n - 3 - k) if k == 1 (mod 2), P(n,k) = min(k - 1, 4n - 2 - k) if k == 0 (mod 2).

A378762 a(n) = A378200(A378200(A378200(n))).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Dec 06 2024

The 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 natural numbers.
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)) = A378200^2(n), a(n) = A378200^3(n), A379342(n) = A378200^4(n), A378705(n) = A378200^5(n). The identity element is A000027(n) = A378200^6(n). - Boris Putievskiy, Jan 15 2025
This sequence, 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, A382680, 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: 2,  3,  6,  5,  4;
  n=3: 9, 10,  7,  8, 15, 14, 13, 12, 11;
(1, 2, 3, ..., 12, 11) = (A378200(1), A378200(2), A378200(3), ..., A378200(14), A378200(15))^3.
(1, 2, 3, ..., 12, 11) = (1, 2, 3, ..., 12, 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,  7, 12, 16, ...
   6, 10, 13, 19, 24, ...
   9, 14, 18, 25, 31, ...
  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;
  3, 4, 1, 2, 9, 8, 7, 6, 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, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

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

Linear sequence: (a(1), a(2), ..., a(A000384(n+1))) is a 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)))^3. (a(1), a(2), ..., a(A000384(n+1))) = (a(1), a(2), ..., a(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 - 1 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, where m = 2n - 1.

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

cp's OEIS Frontend

A378684 a(n) = A378200(A378200(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A379342 a(n) = A378684(A378684(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A378762 a(n) = A378200(A378200(A378200(n))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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.