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

Author

Boris Putievskiy, Feb 24 2024

Keywords

• nonn
• tabf

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

Examples

			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, ...
		

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

Crossrefs

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

Programs

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}]

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.