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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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=2..11.

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

Adjacent sequences:  A278296 A278297 A278298 * A278300 A278301 A278302

KEYWORD

nonn,more

AUTHOR

Alec Jones and Peter Kagey, Nov 17 2016

EXTENSIONS

a(11) from Ryan Lee, May 14 2019

STATUS

approved

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Last modified September 18 10:13 EDT 2021. Contains 347518 sequences. (Running on oeis4.)