A362977 The x-coordinates of the even-ranked elements of the lexicographically earliest sequence of points satisfying staircase and no-three-in-a-line conditions (see comments).

0, 1, 2, 5, 6, 3, 12, 13, 4, 15, 8, 27, 19, 11, 9, 32, 25, 16, 33, 35, 59, 30, 24, 7, 51, 45, 10, 18, 29, 26, 54, 58, 14, 53, 23, 34, 37, 49, 44, 60, 43, 31, 41, 66, 17, 72, 80, 81, 78, 88, 87, 46, 83, 22, 76, 79, 130, 97, 108, 111, 94, 119, 153, 89, 39, 112

Offset

0,3

Comments

We construct the lexicographically earliest sequence of points M(i), for i >= 0, with the following rules:

- their (x, y) coordinates are taken among the nonnegative integers;

- if i > 0 and i is odd, then "move horizontally to a free column"; i.e., M(i) must have the same y as M(i-1), and M(i) is not allowed to have the same x as any M(k) for k < i;

- if i > 0 and i is even, then "move vertically to a free row"; i.e., M(i) must have the same x as M(i-1), and M(i) is not allowed to have the same y as any M(k) for k < i;

- three points are not allowed to be aligned.

See SVG illustration, Links section.

Then a(n) (resp. A362978(n)) is defined as the x-coordinate (resp. y-coordinate) of M(i), where i := 2n (to eliminate duplicates).

Links

Table of n, a(n) for n=0..65.

Luc Rousseau, SVG illustration, n = 0..126

Luc Rousseau, Java program for the combined computation of A362977 and A362978.

Example

  y
  ^
  | . . . 9 . . 8
  | . . 4 . . 5 .
  | . . . . . 6 7
  | . 2 3 . . . .
  | 0 1 . . . . .
  +-------------------> x
    0 1 2 3 4 5 6
Abscissas of the points 0, 2, 4, 6, 8, ...: 0, 1, 2, 5, 6, ...

Prog

(Java) see link.

Crossrefs

Cf. A362978 (the corresponding y-coordinates).

Cf. A161680 (number of nonalignment checks to pass).

Sequence in context: A154925 A154962 A091655 * A353604 A021979 A021043

Adjacent sequences: A362974 A362975 A362976 * A362978 A362979 A362980

Keyword

nonn

Author

Luc Rousseau, May 11 2023

Status

approved