1,7

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 2 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.

Other known values are a(14) = 517344, a(15) = 6068760, a(16) = 339312. a(13) is greater than 800 million.

a(1)..a(6),a(8),a(12) = 0 as these squares cannot be tiled with distinct rectangles with edge lengths that do not divide n. For example for the 8 x 8 square only three rectangles are available with dimensions 3 x 3, 3 x 5, and 5 x 5. All other rectangles have an edge length that divides 8 else leave a space of size 1 or 2 units between its edge and the edge of the square. These gaps cannot be filled as no rectangle can have an edge length of 1 or 2.

a(7) = 360. And example tiling is:

.

+---+---+---+---+---+---+---+

| | | |

+ + + +

| | | |

+---+---+---+---+---+ +

| | |

+ + +

| | |

+---+---+---+---+---+---+---+

| | |

+ + +

| | |

+ + +

| | |

+---+---+---+---+---+---+---+

.

Cf. A360499, A360804, A360256, A360773, A182275, A004003.

Sequence in context: A179678 A101996 A237017 * A097570 A185641 A352225

Adjacent sequences: A360940 A360941 A360942 * A360944 A360945 A360946

nonn,more

Scott R. Shannon, Mar 01 2023

approved