A380200 a(n) = A379343(A379343(n)).

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

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). (a(1), a(2), ..., a(A000384(n+1))) = (A379343(1), A379343(2), ..., A379343(A000384(n+1)))^2. (a(1), a(2), ..., a(A000384(n+1))) = (A379343(1), A379343(2), ..., A379343(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) = k + m if k < m and k == 1 (mod 2), P(n,k) = k - 1 if k < m and k == 0 (mod 2), P(n,k) = k if k >= m and k == 1 (mod 2), P(n, k) = k- m + 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:  5, 2,  4, 3,  6;
  n=3: 12, 7, 14, 9, 11,  8, 13, 10, 15;
(1, 5, 2, ..., 10, 15) = (A379343(1), A379343(2), A379343(3), ..., A379343(14), A379343(15))^2.
(1, 5, 2, ..., 10, 15) = (A379343(1), A379343(2), A379343(3), ..., A379343(14), A379343(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, 14, 10, 27, ...
   4,  7, 13, 18, 26, ...
  12,  8, 25, 19, 42, ...
  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;
  6, 1, 8, 3, 5, 2, 7, 4, 9.

Mathematica

P[n_, k_]:=Module[{m=2*n-1}, If[k<m, If[OddQ[k], k+m, k-1], 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. A000384, A016813 (row lengths), A379343.

Sequence in context: A396610 A111145 A249920 * A380815 A379342 A380245

Adjacent sequences: A380197 A380198 A380199 * A380201 A380202 A380203

Keyword

nonn,tabf

Author

Boris Putievskiy, Jan 16 2025

Status

approved