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

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

1,2

Comments

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

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 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.

Formula

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.

Example

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.

Mathematica

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

Crossrefs

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.

Sequence in context: A254118 A056895 A383723 * A254117 A370629 A218890

Adjacent sequences: A378759 A378760 A378761 * A378763 A378764 A378765

Keyword

nonn,tabf

Author

Boris Putievskiy, Dec 06 2024

Status

approved