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

Author

Luc Rousseau, May 11 2023

Keywords

• nonn

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).

Examples

			  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, ...
		

Links

• Luc Rousseau, SVG illustration, n = 0..126
• Luc Rousseau, Java program for the combined computation of A362977 and A362978.

Crossrefs

Cf. A362978 (the corresponding y-coordinates).

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

Programs

Java

// See Rousseau link.