A379343 Square array read by upward antidiagonals: 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.

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

1,2

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). 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

Example

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;

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}]

Crossrefs

Cf. A016813 (row lengths), A000384.

Sequence in context: A253074 A055266 A338843 * A378684 A381664 A381663

Adjacent sequences: A379340 A379341 A379342 * A379344 A379345 A379346

Keyword

nonn,tabl

Author

Boris Putievskiy, Dec 21 2024

Status

approved