A331621 Number of distinct structures that can be made from n cubes without overhangs.

1, 1, 2, 4, 12, 35, 129, 495, 2101, 9154, 41356, 189466, 880156, 4120515, 19425037, 92038062, 438030079, 2092403558, 10027947217, 48198234188, 232261124908, 1121853426115, 5430222591596

Offset: 0

Author

Nicholas A. Kennedy, Jan 22 2020

Keywords

• nonn
• more

Comments

This is an extension of the free polyominoes (A000105) to the third dimension. Structures are considered equivalent if they can be mapped into each other by reflection in a vertical plane or rotation around the vertical axis. They are not equivalent if they can only be mapped onto each other by rotation around an axis parallel to the horizontal plane.

From John Mason, Mar 03 2025: (Start)

Equivalently, the sequence enumerates inscribed polyominoes that have a positive integer in each square, such that the size of the polyomino is considered to be the sum of the integers.

Examples.

Size 1:

+-+

|1|

+-+

.

Size 2:

+-+-+ +-+

|1|1| |2|

+-+-+ +-+

.

Size 3:

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

|1|1|1| |2|1| |1|1| |3|

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

|1|

+-+

(End)

Examples

			For n = 0, one (the empty) structure is possible.
For n = 1, only one structure is possible, a single cube.
For n = 2, two structures are possible: two cubes one on top of the other, and two next to each other.
For n = 3, four structures are possible: an L shape with the L oriented vertically, an L shape with the L laid flat, a structure with 3 cubes stacked on top of each other and a structure with 3 cubes laid flat.
For n = 4, there are 12 possible distinct structures made from 4 cubes without overhangs. These include 1 structure that is 4 cubes tall, 1 structure that is 3 cubes tall, 5 structures that are 2 cubes tall and 5 that are 1 cube tall.
		

Links

• Arnauld Chevallier, Javascript Node.JS program
• Nicholas A. Kennedy, Buildings made from cubes, Code Golf Stack Exchange 2020
• Nicholas A. Kennedy, Images of possible structures made from 4 cubes
• Miles, Java program

Crossrefs

Extension to the third dimension of A000105.

Cf. A006749, A007318, A259090, A349329, A351616, A234008.

Programs

Java

See Miles link

JavaScript

See Arnauld Chevallier link

Formula

From John Mason, Mar 03 2025: (Start)

Define a(n)=f(n)+g(n) where f(n) enumerates the structures having an asymmetrical base, and g(n) enumerates the structures having a symmetrical base.

Then for n>=4, f(n) = Sum_{i=4..n} ((A006749(i)*C(n-1,i-1)), and g(n) < Sum_{i=1..n} ((A259090(i)*C(n-1,i-1)).

For structures having a base with: reflective orthogonal symmetry about an axis that passes through cell vertices, 180 degree rotational symmetry about a point at a cell vertex or midway along an edge, the number of structures is (1) for odd n: Sum_{i=1..n} ((S(i)*C(n-1,i-1)) / 2, and (2) for even n: Sum_{i=1..n} ((S(i)*(C(n-1,i-1)+C(n/2-1,i/2-1)) / 2, for S(n) = A349329(n), A351616(n) and A234008(n) respectively.

As almost all polyominoes are asymmetrical, a(n)/f(n) tends to 1. (End)

Extensions

a(15)-a(22) from John Mason, Mar 03 2025