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

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

Author

Boris Putievskiy, Apr 03 2025

Keywords

• nonn
• tabf

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

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.

Programs

Mathematica

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

Formula

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.