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A361154 Consider the square grid with cells {(x,y), x, y >= 0}; label the cells by downwards antidiagonals with nonnegative integers so that cells which are a knight's move apart have different labels; always choose smallest possible label.

%I A361154 #37 Mar 09 2023 13:52:14
%S A361154 0,0,0,1,0,1,1,2,2,1,0,1,2,1,0,0,0,2,2,0,0,1,0,3,1,3,0,1,1,1,2,4,4,2,
%T A361154 1,1,0,1,2,3,0,3,2,1,0,0,0,2,2,0,0,2,2,0,0,1,0,3,3,1,0,1,3,3,0,1,1,1,
%U A361154 2,3,1,2,2,1,3,2,1,1,0,1,2,3,0,1,2,1,0,3,2,1,0
%N A361154 Consider the square grid with cells {(x,y), x, y >= 0}; label the cells by downwards antidiagonals with nonnegative integers so that cells which are a knight's move apart have different labels; always choose smallest possible label.
%C A361154 This can also be described as the lexicographically earliest sequence read by downwards antidiagonals in which knight-adjacent cells have distinct labels. [The direction of the diagonals has to be specified, because it can make a difference - as for example if "knight" is replaced by "bishop", when one gets the non-symmetric array A060510.]
%C A361154 Theorem (Spitz): a(n) <= 4. Proof. True at the start, and then by induction, since when labeling a cell there are at most four existing cells that affect it.
%D A361154 Jodi Spitz, Email to _N. J. A. Sloane_, Mar 07 2023
%H A361154 Rémy Sigrist, <a href="/A361154/b361154.txt">Table of n, a(n) for n = 0..10010</a>
%H A361154 Rémy Sigrist, <a href="/A361154/a361154.png">Initial corner of grid showing first 15 antidiagonals</a>. [Different labels have different colors: 0 = red, 1 = orange, etc.]
%H A361154 Rémy Sigrist, <a href="/A361154/a361154_1.png">Initial corner of grid showing cells (x, y) with x, y <= 80</a> [0 = red, 1 = orange, 2 = yellow, 3 = green, 4 = cyan]
%H A361154 Rémy Sigrist, <a href="/A361154/a361154.gp.txt">PARI program</a>
%F A361154 The colors appear to follow an obvious pattern.  For example, the red (0) squares appear to be exactly the squares at (4*i + d, 4*j + e), for i and j >= 0, d and e = 0 or 1. The blue (4) squares appear to be exactly the squares at (4*k, 4*k - 1) and (4*k - 1, 4*k), for k >= 1. - _N. J. A. Sloane_, Mar 07 2023
%e A361154 The initial antidiagonals are:
%e A361154   0,
%e A361154   0, 0,
%e A361154   1, 0, 1,
%e A361154   1, 2, 2, 1,
%e A361154   0, 1, 2, 1, 0,
%e A361154   0, 0, 2, 2, 0, 0,
%e A361154   1, 0, 3, 1, 3, 0, 1,
%e A361154   1, 1, 2, 4, 4, 2, 1, 1,
%e A361154   0, 1, 2, 3, 0, 3, 2, 1, 0,
%e A361154   0, 0, 2, 2, 0, 0, 2, 2, 0, 0,
%e A361154   1, 0, 3, 3, 1, 0, 1, 3, 3, 0, 1,
%e A361154   1, 1, 2, 3, 1, 2, 2, 1, 3, 2, 1, 1,
%e A361154   0, 1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 1, 0,
%e A361154 ...
%o A361154 (PARI) See Links section.
%Y A361154 Cf. A060510, A308884.
%K A361154 nonn,tabl
%O A361154 0,8
%A A361154 _N. J. A. Sloane_, Mar 07 2023, based on an email from _Jodi Spitz_, Mar 07 2023
%E A361154 Data corrected by _Rémy Sigrist_, Mar 07 2023