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A361154
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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.
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1
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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, 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, 2, 3, 1, 2, 2, 1, 3, 2, 1, 1, 0, 1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 1, 0
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OFFSET
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0,8
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COMMENTS
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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.]
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.
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REFERENCES
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LINKS
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FORMULA
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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
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EXAMPLE
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The initial antidiagonals are:
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, 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, 2, 3, 1, 2, 2, 1, 3, 2, 1, 1,
0, 1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 1, 0,
...
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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