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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.

%I A278299 #32 May 14 2019 19:22:12
%S A278299 2,4,6,9,12,15,19,24,30,34
%N 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.
%C A278299 Only edge-to-edge adjacencies are considered.
%C A278299 The sequence is bounded above by A053439(n-1).
%C A278299 a(n) is bounded below by n * ceiling((n - 1)/4). This bound is achieved for n=2, n=6, and n=10.
%e A278299 Example: for n = 4, the following diagram gives a minimal polyomino of a(4) = 6 tiles:
%e A278299       +---+---+
%e A278299       | 1 | 4 |
%e A278299   +---+---+---+
%e A278299   | 4 | 3 | 2 |
%e A278299   +---+---+---+
%e A278299           | 1 |
%e A278299           +---+
%e A278299 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.
%e A278299               +---+---+
%e A278299               | 8 | 6 |
%e A278299           +---+---+---+---+---+
%e A278299           | 3 | 2 | 5 | 9 | 4 |
%e A278299   +---+---+---+---+---+---+---+---+
%e A278299   | 2 | 7 | 5 | 1 | 4 | 2 | 10| 9 |
%e A278299   +---+---+---+---+---+---+---+---+
%e A278299   | 6 | 9 | 8 | 3 | 6 | 7 | 8 | 1 |
%e A278299   +---+---+---+---+---+---+---+---+
%e A278299   | 10| 3 | 4 | 7 | 1 | 10| 5 |
%e A278299   +---+---+---+---+---+---+---+
%e A278299 From _Ryan Lee_, May 14 2019: (Start)
%e A278299 Example for n = 11:
%e A278299   +---+---+---+---+---+
%e A278299   | 9 | 11| 2 | 5 | 8 |
%e A278299   +---+---+---+---+---+---+
%e A278299   | 1 | 5 | 10| 9 | 2 | 1 |
%e A278299   +---+---+---+---+---+---+
%e A278299   | 4 | 6 | 11| 8 | 7 | 3 |
%e A278299   +---+---+---+---+---+---+
%e A278299   | 3 | 9 | 7 | 10| 6 | 2 |
%e A278299   +---+---+---+---+---+---+
%e A278299   | 11| 4 | 5 | 3 | 8 | 4 |
%e A278299   +---+---+---+---+---+---+
%e A278299   | 1 | 10|   | 6 | 1 | 7 |
%e A278299   +---+---+   +---+---+---+
%e A278299 (End)
%Y A278299 Cf. A053439.
%K A278299 nonn,more
%O A278299 2,1
%A A278299 _Alec Jones_ and _Peter Kagey_, Nov 17 2016
%E A278299 a(11) from _Ryan Lee_, May 14 2019