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

1,6

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

Links

Table of n, a(n) for n=1..16.

Rodolfo Kurchan, One Up Puzzle.

Rodolfo Kurchan, Puzzle Fun (see One Up).

Example

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.

Crossrefs

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

Sequence in context: A031453 A345961 A179203 * A102017 A032916 A044994

Adjacent sequences: A371473 A371474 A371475 * A371477 A371478 A371479

Keyword

nonn,more

Author

Rodolfo Kurchan, Mar 24 2024

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

Status

approved