A379343 Square array read by upward antidiagonals: T(n,k) = (2*(k+n-1)^2 + 2*k - 2*n + 3 + (-2*k - 2*n + 3)*(-1)^n - (-1)^k + (-2*k - 2*n + 1)*(-1)^(k+n))/4.

1, 3, 5, 4, 2, 6, 8, 12, 10, 14, 11, 7, 13, 9, 15, 17, 23, 19, 25, 21, 27, 22, 16, 24, 18, 26, 20, 28, 30, 38, 32, 40, 34, 42, 36, 44, 37, 29, 39, 31, 41, 33, 43, 35, 45, 47, 57, 49, 59, 51, 61, 53, 63, 55, 65, 56, 46, 58, 48, 60, 50, 62, 52, 64, 54, 66, 68, 80, 70, 82, 72, 84, 74, 86, 76, 88, 78, 90, 79, 67, 81, 69, 83, 71, 85, 73, 87, 75, 89, 77

Offset: 1

Author

Boris Putievskiy, Dec 21 2024

Keywords

• nonn
• tabl
• easy

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.

This sequence and A378684 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the alternating group A4. The list of the 12 elements of that group: this sequence, A380200 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A378684, A379342, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Mar 11 2025

Examples

			Square array begins:
  1,  5,  6, 14, 15, ...
  3,  2, 10,  9, 21, ...
  4, 12, 13, 25, 26, ...
  8,  7, 19, 18, 34, ...
 11, 23, 24, 40, 41, ...
  ...
The first 5 antidiagonals are:
  1;
  3,  5;
  4,  2,  6;
  8, 12, 10, 14;
 11,  7, 13,  9, 15;
Triangle array begins:
  k=    1   2   3   4   5   6   7   8   9
  n=1:  1;
  n=2:  3,  5,  4,  2,  6;
  n=3:  8, 12, 10, 14, 11,  7, 13,  9, 15;
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table.
ord(1,3,5... 9,15) = 3.
Subtracting (n-1)*(2*n-3) from each term is row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  2, 4, 3, 1, 5;
  2, 6, 4, 8, 5, 1, 7, 3, 9;
		

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.
• Boris Putievskiy, The Alternating Group A4: Subgroups and the Cayley Table (2025).
• Eric Weisstein's World of Mathematics, Alternating Group.
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A016813 (row lengths), A000384, A000027, A381662, A380817, A376214, A380200, A378684, A379342, A380245, A381664, A380815, A381663.

Programs

Mathematica

T[n_,k_]:=(2*(k+n-1)^2+2k-2n+3+(-2k-2n+3)*(-1)^n-(-1)^k+(-2k-2n+1)*(-1)^(k+n))/4;Table[T[k,n],{k,1,5},{n,1,5}]

Formula

Linear sequence: (a(1),a(2), ... a(A000384(n+1))) is a 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 + 1, 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, P(n,k) = k, if k => m(n) and k mod 2 = 1, P(n,k) = k - m(n), if k => m(n) and k mod 2 = 0, where m(n) = 2n-1.

G.f.: x*y*(1 + x^8*y^4 + x*(2 + y) + x^7*y^3*(9 + 8*y) - x^3*y*(8 + 8*y - 3*y^2) - x^2*(1 - 8*y - 9*y^2) + x^6*(y^2 - 4*y^3 - 5*y^4) + x^5*(3*y - 8*y^3 - 2*y^4) + 2*x^4*(1 - 2*y - 5*y^2 + y^4))/((1 - x)^3*(1 + x)^2*(1 - x*y)^3*(1 + x*y)^2). - Stefano Spezia, Dec 21 2024