A393939 Number of free polycubes with n cells such that pairs of cell centers determine the largest possible number (A394394(n)) of distinct lines.

1, 1, 1, 4, 5, 18, 33, 67, 82, 129, 190, 278, 365, 450, 483, 479, 455, 422, 356, 284, 208, 138, 72, 28, 8, 1, 1235, 469, 118, 21, 2

Offset

1,4

Comments

For 1 <= n <= 26, a(n) is the number of free polycubes with n cells and no 3 collinear cell centers, i.e., pairs of cell centers determine n*(n-1)/2 distinct lines. No such polycube exists for n > 26.

For 27 <= n <= 31, a(n) is the number of free polycubes with 3 cell centers on a line but no other collinearities, i.e., pairs of cell centers determine n*(n-1)/2-2 distinct lines. No such polycube exists for n > 31.

The unique free polycube with 26 cells and no 3 collinear cells (see linked illustration): {(0,1,2), (0,1,3), (0,2,1), (0,2,2), (1,0,3), (1,1,3), (1,1,4), (1,2,1), (1,2,4), (1,3,1), (1,3,2), (1,4,2), (2,0,2), (2,0,3), (2,1,0), (2,1,1), (2,2,4), (2,2,5), (2,3,3), (2,3,4), (3,0,2), (3,1,1), (3,1,2), (3,2,3), (3,3,3), (4,2,3)}.

Links

Table of n, a(n) for n=1..31.

Pontus von Brömssen, Illustration of the unique free polycube with 26 cells and no 3 collinear cells.

Index entries for sequences related to polyominoes.

Formula

a(n) = A393938(n,1) + A393938(n,2) for 1 <= n <= 26.

Crossrefs

Cf. A038119, A378169, A393938, A394392 (polyominoes), A394394, A394395 (distinct directions of lines).

Sequence in context: A357366 A026902 A026755 * A368029 A243120 A317378

Adjacent sequences: A393936 A393937 A393938 * A393940 A393941 A393942

Keyword

nonn,more

Author

Pontus von Brömssen, Mar 04 2026

Status

approved