A371476 - OEIS

%I #34 Apr 11 2024 17:22:26

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

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

%C 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

%H Rodolfo Kurchan, <a href="https://www.oneuppuzzle.com/">One Up Puzzle</a>.

%H Rodolfo Kurchan, <a href="https://www.puzzlefun.online/problems">Puzzle Fun</a> (see One Up).

%e The a(6) = 3 solvable hexominoes have unique solutions:

%e   +---+           +---+---+       +---+---+

%e   | 1 |           | 1 | 2 |       | 2 | 1 |

%e   +---+---+       +---+---+---+   +---+---+---+

%e   | 2 | 1 |       | 2 | 3 | 1 |   | 3 | 2 | 1 |

%e   +---+---+---+   +---+---+---+   +---+---+---+

%e   | 3 | 2 | 1 |       | 1 |       | 1 |

%e   +---+---+---+       +---+       +---+

%e a(7) = 10 because there are 10 heptominoes that have at least one solution to the One Up puzzle.

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

%K nonn,more

%O 1,6

%A _Rodolfo Kurchan_, Mar 24 2024

%E a(7)-a(14) from _Pontus von Brömssen_, Mar 25 2024

%E a(15) from _Pontus von Brömssen_, Mar 26 2024

%E a(16) from _Pontus von Brömssen_, Apr 04 2024