A371476 a(n) is the number of free polyominoes of size n with at least one solution to the One Up puzzle (see comments).

1, 0, 1, 1, 1, 3, 10, 12, 23, 35, 169, 255, 817, 1883, 4702, 10489

Offset: 1

Author

Rodolfo Kurchan, Mar 24 2024

Keywords

• nonn
• more

Comments

The objective of the One Up puzzle is to assign a positive integer to each cell of a given polyomino in such a way that the cells of any maximal 1 X k strip (horizontal or vertical) are numbered 1, ..., k (in some order). The maximality is applied to horizontal and vertical strips separately, implying that the number 1 must be assigned to a cell with no left or right neighbors even if it has neighbors above or below (and vice versa). (In an extended version of the puzzle, there may be walls between certain pairs of neighboring cells, and only those strips that do not extend over a wall are considered.) - Pontus von Brömssen, Mar 26 2024

Examples

			The a(6) = 3 solvable hexominoes have unique solutions:
  +---+           +---+---+       +---+---+
  | 1 |           | 1 | 2 |       | 2 | 1 |
  +---+---+       +---+---+---+   +---+---+---+
  | 2 | 1 |       | 2 | 3 | 1 |   | 3 | 2 | 1 |
  +---+---+---+   +---+---+---+   +---+---+---+
  | 3 | 2 | 1 |       | 1 |       | 1 |
  +---+---+---+       +---+       +---+
a(7) = 10 because there are 10 heptominoes that have at least one solution to the One Up puzzle.
		

Links

• Rodolfo Kurchan, One Up Puzzle.
• Rodolfo Kurchan, Puzzle Fun (see One Up).

Crossrefs

Cf. A000105, A371828 (a generalization to hypergraphs).

Extensions

a(7)-a(14) from Pontus von Brömssen, Mar 25 2024

a(15) from Pontus von Brömssen, Mar 26 2024

a(16) from Pontus von Brömssen, Apr 04 2024