cp's OEIS Frontend

The sequence gives the number of distinct tilings by polyominoes of a square with side n, considering tilings that are formed of distinct polyominoes. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

The sequence gives the number of distinct tilings by polyominoes of a square with side n, considering tilings that are formed by polyominoes of all different sizes. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

The sequence gives the number of distinct tilings by polyominoes of a square with side n, considering tilings that have at least one symmetry. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

The areas of size 1 through n can be created in any order and position, the only requirement being the final number of line segments used to enclose all areas is minimized. It is likely the perimeter of each area of size k, 1 <= k <= n, is the minimum possible for an area of size k, although this is unknown.

See A348149 for the total segments when the number of segments at each step is minimized.

In this variation of A347581 the areas must be added in the order of their sizes, from 1 through n, and as each area is added the minimum possible number of line segments must be used. This forces, for example, the first three areas of size 1, 2 and 3 to form a 2 X 3 block and thus they can never appear in any other arrangement in the final area. This is also true for n up to at least 9 due to the restriction of maximizing the usable edges for the next area. This leads to a(8) and a(10) containing one more line segment than the optimal solutions of A347581.

Consider n unit squares contained within an n X n square. The n unit squares are an n-cell pattern of the n X n square if, when copied n-1 times, they can, with various rigid transformations, be combined to tessellate the n X n square.

Put another way:

Consider, for example, for n = 4, a transparency with an outline of a 4 X 4 square filled by 16 unit squares. Any 4 unit squares are painted the same color. Those four squares are a potential n-cell pattern of the 4 X 4 square. Three copies of the transparency are made with only the color of the 4 squares being different. If a person can stack the transparencies in such a way that they fill the 4 X 4 square, then the n-cell pattern is acceptable.

Put another way:

n unit squares from an n X n square are painted a color. Those n unit squares are an n-cell pattern. If n-1 copies of the pattern can be painted (each a different color) and together they fill the n X n square, then the n unit squares form an acceptable n-cell pattern.

Conjecture by Andrew Young: For an n X n square, where n is an odd prime, there is only one n-cell pattern.

Conjecture by Andrew Young and Thomas Young: An odd integer n>=3 is prime iff there exists only one n-cell pattern for an n X n square.

For any number n, there is always the 1 X n pattern that tiles the n X n square.

For composite numbers n = f1*f2, 1 < f1 <= f2 < n, there is always an additional f1 X f2 pattern. For example, a 14 X 14 square can be tiled using fourteen 1 X 14 rectangles or fourteen 2 X 7 rectangles; a 15 X 15 square can be tiled using fifteen 1 X 15 rectangles or fifteen 3 X 5 rectangles; a 9 X 9 square can be tiled using nine 1 X 9 rectangles or nine 3 X 3 squares (as in Sudoku!).

The second conjecture is a Corollary of the first: If n = p*q is not prime, then there is always a second tiling using rectangles, as explained above. Since the second conjecture implies the first, the two conjectures are actually equivalent. - M. F. Hasler, Jun 15 2025