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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.
1
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
OFFSET
0,8
COMMENTS
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.
REFERENCES
Jodi Spitz, Email to N. J. A. Sloane, Mar 07 2023
LINKS
Rémy Sigrist, Initial corner of grid showing first 15 antidiagonals. [Different labels have different colors: 0 = red, 1 = orange, etc.]
Rémy Sigrist, Initial corner of grid showing cells (x, y) with x, y <= 80 [0 = red, 1 = orange, 2 = yellow, 3 = green, 4 = cyan]
Rémy Sigrist, PARI program
FORMULA
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
EXAMPLE
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,
...
PROG
(PARI) See Links section.
CROSSREFS
Sequence in context: A107063 A290453 A108423 * A208568 A193804 A180424
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Mar 07 2023, based on an email from Jodi Spitz, Mar 07 2023
EXTENSIONS
Data corrected by Rémy Sigrist, Mar 07 2023
STATUS
approved