A383723 a(n) = A378762(A376214(n)).

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

Offset

1,2

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.

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

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = m - 1 - k if k < m and k == 1 (mod 2), P(n, k) = k if k < m and k == 0 (mod 2), P(n, k) = 3m - 1 - k if k >= m and k == 1 (mod 2), P(n, k) = k 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:  2, 3, 6,  5,  4;
  n=3:  9, 8, 7, 10, 15, 12, 13, 14, 11;
(1, 2, 3, ..., 14, 11) = (1, 2, 3, ..., 12, 11) (1, 2, 3, ..., 12, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A376214.
(1, 2, 3, ..., 14, 11) = (1, 2, 3, ..., 14, 11)^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4, 10, 11, ...
   2,  5,  7, 14, 16, ...
   6,  8, 13, 19, 24, ...
   9, 12, 18, 25, 31, ...
  15, 17, 26, 32, 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, 2, 1, 4, 9, 6, 7, 8, 5.

Mathematica

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

Crossrefs

Cf. A000027, A000384, A016813 (row lengths), A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680.

Sequence in context: A384564 A254118 A056895 * A378762 A254117 A370629

Adjacent sequences: A383720 A383721 A383722 * A383724 A383725 A383726

Keyword

nonn,tabf

Author

Boris Putievskiy, May 07 2025

Status

approved