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

1,2

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 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

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.

Index entries for sequences that are permutations of the natural numbers.

Formula

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 = 2n - 1.

Example

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.

Mathematica

P[n_, k_]:=Module[{m=2*n-1}, If[k<m, If[OddQ[k], k+1, 2*m-k], If[OddQ[k], k, 2*m-k-1]]]
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), A370655, A373498, A374447, A374494, A374531, A375602, A375725, A378200, A378705, A378762, A379342.

Sequence in context: A055266 A338843 A379343 * A381664 A381663 A075077

Adjacent sequences: A378681 A378682 A378683 * A378685 A378686 A378687

Keyword

nonn,tabf,changed

Author

Boris Putievskiy, Dec 03 2024

Status

approved