A378200 Table T(n, k) read by upward antidiagonals. T(n,k) = ((k + n - 1)^2 + (k - n + 1)(-1)^n + (1 - k - n)(-1)^k + (1 - k - n)(-1)^(k + n) + 2) / 2.

1, 5, 2, 6, 3, 4, 12, 9, 14, 7, 15, 8, 13, 10, 11, 23, 20, 25, 18, 27, 16, 28, 17, 26, 19, 24, 21, 22, 38, 35, 40, 33, 42, 31, 44, 29, 45, 30, 43, 32, 41, 34, 39, 36, 37, 57, 54, 59, 52, 61, 50, 63, 48, 65, 46, 66, 47, 64, 49, 62, 51, 60, 53, 58, 55, 56, 80, 77, 82, 75, 84, 73, 86, 71, 88, 69, 90, 67, 91, 68, 89, 70, 87, 72, 85, 74, 83, 76, 81, 78

Offset

1,2

Comments

The sequence can be arranged in a triangular array, read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n has a length of 4n-3 = A016813(n+1), n > 0. For n > 1, each row is a pair 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..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

Triangular array T(n,k) for 1 <= k <= 4n - 3 (see Example):

T(n,k) = A000384(n-1) + P(n,k), P(n,k) = m(n) + k, if k < m(n) and k mod 2 = 1,

P(n,k) = m(n) - k , if k < m(n) and k mod 2 = 0, P(n,k) = 3m(n) - k - 1, if k => m(n) and k mod 2 = 1, P(n,k)= - m(n) + k + 1, if k => m(n) and k mod 2 = 0, where m(n) = 2n-1.

Example

Table begins:
  1,  2,  4,  7, 11, ...
  5,  3, 14, 10, 27, ...
  6,  9, 13, 18, 24, ...
 12,  8, 25, 19, 42, ...
 15, 20, 26, 33, 41, ...
 ...
The first 5 antidiagonals are:
  1;
  5, 2;
  6, 3,  4;
 12, 9, 14, 7;
 15, 8, 13,10, 11;
Triangle array begins:
  k=   1   2   3   4   5   6   7   8   9
  n=1: 1;
  n=2: 5,  2,  6,  3,  4;
  n=3: 12, 9, 14,  7,  15, 8, 13, 10, 11;
Subtracting (n-1)*(2*n-3) from each term is row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  4, 1, 5, 2, 3;
  6, 3, 8, 1, 9, 2, 7, 4, 5;

Mathematica

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

Crossrefs

Cf. A016813 (row lengths), A000384, A370655, A373498, A374447, A374494, A374531, A375602, A375725.

Sequence in context: A152025 A021099 A021023 * A078716 A308081 A164103

Adjacent sequences: A378197 A378198 A378199 * A378201 A378202 A378203

Keyword

nonn,tabl

Author

Boris Putievskiy, Nov 19 2024

Status

approved