A380245 a(n) = A379343(A378684(n)).

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

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 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.

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 + 2*m(n) - 1 if k < m(n) and k mod 2 = 1, P(n, k) = m(n) - k if k < m(n) and k mod 2 = 0, P(n, k) = k if k >= m(n) and k mod 2 = 1, P(n, k) = k - m(n) + 1 if k >= m(n) and k mod 2 = 0, where m(n) = 2*n - 1.

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, 9, 12, 7, 11,  8, 13, 10, 15;
ord(1,5,2... 10,15) = 3.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
  1,  2,  6,  7, 15, ...
  5,  3, 12, 10, 23, ...
  4,  9, 13, 18, 26, ...
 14,  8, 25, 19, 40, ...
 11, 20, 24, 33, 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, 3, 6, 1, 5, 2, 7, 4, 9;

Mathematica

P[n_, k_]:=Module[{m=2*n-1}, If[k<m, If[OddQ[k], -k+2m-1, m-k], If[OddQ[k], k, k-m+1]]]
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, A378684, A379343.

Sequence in context: A249920 A380200 A379342 * A021660 A266122 A064853

Adjacent sequences: A380242 A380243 A380244 * A380246 A380247 A380248

Keyword

nonn,tabf

Author

Boris Putievskiy, Jan 17 2025

Status

approved