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

Offset

0,3

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.

Links

Table of n, a(n) for n=0..14.

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

Example

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.

Prog

(JavaScript) See Arnauld Chevallier link
(Java) See Miles link

Crossrefs

Extension to the third dimension of A000105.

Sequence in context: A019447 A148206 A148207 * A334250 A211768 A112083

Adjacent sequences: A331618 A331619 A331620 * A331622 A331623 A331624

Keyword

nonn,more

Author

Nicholas A. Kennedy, Jan 22 2020

Status

approved