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A278299
a(n) is the tile count of the smallest polyomino with an n-coloring such that every color is adjacent to every other distinct color at least once.
0
2, 4, 6, 9, 12, 15, 19, 24, 30, 34
OFFSET
2,1
COMMENTS
Only edge-to-edge adjacencies are considered.
The sequence is bounded above by A053439(n-1).
a(n) is bounded below by n * ceiling((n - 1)/4). This bound is achieved for n=2, n=6, and n=10.
EXAMPLE
Example: for n = 4, the following diagram gives a minimal polyomino of a(4) = 6 tiles:
+---+---+
| 1 | 4 |
+---+---+---+
| 4 | 3 | 2 |
+---+---+---+
| 1 |
+---+
Example: for n = 10, the following diagram gives a minimal polyomino of a(10) = 30 tiles. Note that redundant adjacencies, e.g., between 2 and 7, can exist in minimal diagrams.
+---+---+
| 8 | 6 |
+---+---+---+---+---+
| 3 | 2 | 5 | 9 | 4 |
+---+---+---+---+---+---+---+---+
| 2 | 7 | 5 | 1 | 4 | 2 | 10| 9 |
+---+---+---+---+---+---+---+---+
| 6 | 9 | 8 | 3 | 6 | 7 | 8 | 1 |
+---+---+---+---+---+---+---+---+
| 10| 3 | 4 | 7 | 1 | 10| 5 |
+---+---+---+---+---+---+---+
From Ryan Lee, May 14 2019: (Start)
Example for n = 11:
+---+---+---+---+---+
| 9 | 11| 2 | 5 | 8 |
+---+---+---+---+---+---+
| 1 | 5 | 10| 9 | 2 | 1 |
+---+---+---+---+---+---+
| 4 | 6 | 11| 8 | 7 | 3 |
+---+---+---+---+---+---+
| 3 | 9 | 7 | 10| 6 | 2 |
+---+---+---+---+---+---+
| 11| 4 | 5 | 3 | 8 | 4 |
+---+---+---+---+---+---+
| 1 | 10| | 6 | 1 | 7 |
+---+---+ +---+---+---+
(End)
CROSSREFS
Cf. A053439.
Sequence in context: A194203 A261222 A130025 * A145802 A076271 A036441
KEYWORD
nonn,more
AUTHOR
Alec Jones and Peter Kagey, Nov 17 2016
EXTENSIONS
a(11) from Ryan Lee, May 14 2019
STATUS
approved