A379342 a(n) = A378684(A378684(n)).

1, 5, 2, 4, 3, 6, 14, 7, 12, 9, 11, 10, 13, 8, 15, 27, 16, 25, 18, 23, 20, 22, 21, 24, 19, 26, 17, 28, 44, 29, 42, 31, 40, 33, 38, 35, 37, 36, 39, 34, 41, 32, 43, 30, 45, 65, 46, 63, 48, 61, 50, 59, 52, 57, 54, 56, 55, 58, 53, 60, 51, 62, 49, 64, 47, 66, 90, 67, 88, 69, 86, 71, 84, 73, 82, 75, 80, 77, 79, 78, 81, 76, 83, 74

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.

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 a permutation of the positive integers from 1 to A000384(n+1). ord((a(1),a(2), ... a(A000384(n+1)) = 3, where ord is the order of the permutation. (a(1),a(2), ... a(A000384(n+1)) = (A378684(1), A378684(2), ... A378684(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) = max(k, 4n - 3 - k) if k mod 2 = 1, P(n, k) = min(k - 1, 4n - 2 - k) if k mod 2 = 0.

Example

Triangle array begins:
  k=   1   2   3   4   5   6   7   8   9
  n=1: 1;
  n=2: 5,  2,  4,  3,  6;
  n=3: 14, 7, 12,  9, 11, 10, 13,  8, 15;
  ...
(1,5,2, ...8,15) = (A378684(1), A378684(2),  A378684(3), ... A378684(14), A378684(15))^2. (1,5,2, ...8,15) = (A378684(1), A378684(2),  A378684(3), ... A378684(14), A378684(15))^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
  1,  2,  6,  9, 15, ...
  5,  3, 12,  8, 23, ...
  4,  7, 13, 18, 26, ...
 14, 10, 25, 19, 40, ...
 11, 16, 24, 31, 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, 1, 3, 2, 5;
  8, 1, 6, 3, 5, 4, 7, 2, 9;

Mathematica

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

Crossrefs

Cf. A016813 (row lengths), A000384, A370655, A373498, A374447, A374494, A374531, A375469, A375602, A375725, A378200.

Sequence in context: A083241 A111145 A249920 * A021660 A266122 A064853

Adjacent sequences: A379339 A379340 A379341 * A379344 A379345 A379346

Keyword

nonn,tabf,new

Author

Boris Putievskiy, Dec 21 2024

Status

approved