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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).

%I A362977 #26 Jan 06 2025 21:46:04
%S A362977 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,
%T A362977 10,18,29,26,54,58,14,53,23,34,37,49,44,60,43,31,41,66,17,72,80,81,78,
%U A362977 88,87,46,83,22,76,79,130,97,108,111,94,119,153,89,39,112
%N 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).
%C A362977 We construct the lexicographically earliest sequence of points M(i), for i >= 0, with the following rules:
%C A362977 - their (x, y) coordinates are taken among the nonnegative integers;
%C A362977 - 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;
%C A362977 - 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;
%C A362977 - three points are not allowed to be aligned.
%C A362977 See SVG illustration, Links section.
%C A362977 Then a(n) (resp. A362978(n)) is defined as the x-coordinate (resp. y-coordinate) of M(i), where i := 2n (to eliminate duplicates).
%H A362977 Luc Rousseau, <a href="/A362977/a362977.svg">SVG illustration, n = 0..126</a>
%H A362977 Luc Rousseau, <a href="/A362977/a362977.java.txt">Java program for the combined computation of A362977 and A362978.</a>
%e A362977   y
%e A362977   ^
%e A362977   | . . . 9 . . 8
%e A362977   | . . 4 . . 5 .
%e A362977   | . . . . . 6 7
%e A362977   | . 2 3 . . . .
%e A362977   | 0 1 . . . . .
%e A362977   +-------------------> x
%e A362977     0 1 2 3 4 5 6
%e A362977 Abscissas of the points 0, 2, 4, 6, 8, ...: 0, 1, 2, 5, 6, ...
%o A362977 (Java) // See Rousseau link.
%Y A362977 Cf. A362978 (the corresponding y-coordinates).
%Y A362977 Cf. A161680 (number of nonalignment checks to pass).
%K A362977 nonn
%O A362977 0,3
%A A362977 _Luc Rousseau_, May 11 2023