A373498 a(a(a(n))) = A370655(n).

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

Offset: 1

Author

Boris Putievskiy, Jun 17 2024

Keywords

• nonn
• tabf

Comments

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.

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

The sequence is an intra-block permutation of positive integers.

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 02 2024

Examples

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   2,  1,  3;
  n=2:   5,  9,  7,  6,  8,  4, 10;
  n=3:  12, 18, 14, 20, 16, 15, 17, 13, 19, 11, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
   2,  3,  7, 10, 16, 21, ...
   1,  9,  4, 20, 11, 35, ...
   5,  8, 14, 19, 27, 34, ...
   6, 18, 13, 33, 24, 52, ...
  12, 17, 25, 32, 42, 51, ...
  15, 31, 26, 50, 41, 73, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
   2,1,3,
   2,6,4,3,5,1,7,
   2,8,4,10,6,5,7,3,9,1,11,
   ...
The 3rd power of each permutation is equal to the corresponding permutation in example A370655:
   (2,1,3)^3 = (2,1,3),
   (2,6,4,3,5,1,7)^3 = (1,2,4,3,5,6,7),
   (2,8,4,10,6,5,7,3,9,1,11)^3 = (3,4,1,2,6,5,7,10,9,8,11).
		

Links

• Boris Putievskiy, Table of n, a(n) for n = 1..9870 _Boris Putievskiy_, Aug 02 2024
• 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
• _Boris Putievskiy_, Aug 02 2024

Crossrefs

Cf. A000027, A004767 (row lengths), A204164, A370655, 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=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}]
Table[a[a[a[n]]],{n,1,Nmax}] (* A370655 *)

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) + 1 if R(n) <= 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 2*L(n) + R(n) if R(n) <= 2*L(n)-1 and R(n) mod 2 = 0, P(n) = R(n) if R(n) > 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 4*L(n) - R(n) - 1 if R(n) > 2*L(n)-1 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) = k + 1 if k <= 2*n-1 and k mod 2 = 1, P(n,k) = 2*n + k if k <= 2*n-1 and k mod 2 = 0,

P(n,k) = k if k > 2*n-1 and k mod 2 = 1, P(n,k) = 4*n - k - 1 if k > 2*n-1 and k mod 2 = 0.