A370655 Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.

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

Offset

1,1

Comments

Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.

The sequence is a self-inverse permutation of natural numbers.

The sequence is an intra-block permutation of integer positive numbers.

The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself: A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6. - Boris Putievskiy, Aug 03 2024

Links

Boris Putievskiy, Table of n, a(n) for n = 1..9870

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(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),

L(n) = A204164(n),

R(n) = n - (L(n)-1)*(2*L(n)-1),

P(n) = -R(n) + 2*L(n)-2, if R(n) < 2*L(n) - 1 and R(n) mod 2 = 1, P(n) = -R(n) + 2*L(n), if R(n) < 2*L(n) - 1 and R(n) mod 2 = 0, P(n) = 2*L(n), if R(n) = 2*L(n) - 1, P(n) = R(n)-1, if R(n) = 2*L(n), P(n) = R(n), if R(n) > 2*L(n) and R(n) mod 2 = 1, P(n) = 6*L(n) - R(n), if R(n) > 2*L(n) and R(n) mod 2 = 0.

Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):

T(n,k) = (n-1)*(2*n-1) + P(n,k), where

P(n,k) = 2*n-k-2 if k < 2*n-1 and k mod 2 = 1,

2*n-k if k < 2*n-1 and k mod 2 = 0,

2*k if k = 2*n-1,

k-1 if k = 2*n,

k if k > 2*n and k mod 2 = 1,

6*n-k if k > 2*n and k mod 2 = 0.

Example

Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   2,  1,  3;
  n=2:   4,  5,  7,  6,  8,  9, 10;
  n=3:  13, 14, 11, 12, 16, 15, 17, 20, 19, 18, 21;
Subtracting (n-1)*(2*n-1) from each term is row n is a self-inverse permutation of 1 .. 4*n-1,
  2,1,3,
  1,2,4,3,5,6,7,
  3,4,1,2,6,5,7,10,9,8,11,
  ...
The triangle rows can be arranged as two successive upward antidiagonals in an array:
   2,  3,  7, 10, 16, 21, ...
   1,  5,  9, 12, 18, 23, ...
   4,  8, 11, 19, 22, 34, ...
   6, 14, 20, 25, 33, 40, ...
  13, 17, 24, 32, 39, 51, ...
  15, 27, 35, 42, 52, 61, ...

Mathematica

Nmax = 21;
a[n_] := Module[{L, R, P, Result}, L = Ceiling[(Sqrt[8*n + 1] - 1)/4];
  R = n - (L - 1)*(2*L - 1);
  P = If[R < 2*L - 1, If[Mod[R, 2] == 1, -R + 2*L - 2, -R + 2*L],
    If[R == 2*L - 1, 2*L,
     If[R == 2*L, R - 1, If[Mod[R, 2] == 1, R, 6*L - R]]]];
  Result = P + (L - 1)*(2*L - 1);
  Result]
Table[a[n], {n, 1, Nmax}]

Crossrefs

Cf. A000027, A004767 (row lengths), A204164, A373498, A374447, A374494, A374531.

Sequence in context: A346786 A239469 A282840 * A082750 A048212 A282864

Adjacent sequences: A370652 A370653 A370654 * A370656 A370657 A370658

Keyword

nonn,tabf

Author

Boris Putievskiy, Feb 24 2024

Status

approved