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

Author

Boris Putievskiy, Dec 03 2024

Keywords

• nonn
• tabf

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

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

Examples

			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.
		

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 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.

Crossrefs

Cf. A000027, A000384, A016813 (row lengths), A376214, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664.

Programs

Mathematica

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

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 = 2*n - 1.