A360632 Number of sets of integer-sided rectangular pieces that can tile a 3 X n rectangle.

1, 3, 10, 21, 73, 190, 510, 1196, 2895, 6437, 14281, 29840, 62405, 124506, 246383, 473094, 899000, 1665763, 3057894, 5500889, 9808150

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Example

From Robin Visser, May 01 2025: (Start)
For n = 1, there are a(1) = 3 possible sets of rectangular pieces that can tile a 3 x 1 rectangle: one 1 x 3 piece; one 1 x 2 piece and one 1 x 1 piece; or three 1 x 1 pieces.
For n = 2, there are a(2) = 10 possible sets of rectangular pieces that can tile a 3 x 2 rectangle: one 2 x 3 piece; one 2 x 2 piece and one 1 x 2 piece; one 2 x 2 piece and two 1 x 1 pieces; two 1 x 3 pieces; one 1 x 3 piece, one 1 x 2 piece, and one 1 x 1 piece; one 1 x 3 piece and three 1 x 1 pieces; three 1 x 2 pieces; two 1 x 2 pieces and two 1 x 1 pieces; one 1 x 2 piece and four 1 x 1 pieces; or six 1 x 1 pieces. (End)

Prog

(Python)
def a(n):
    A = [[[set() for i in range(n+1)] for j in range(n+1)] for k in range(n+1)]
    A[0][0][0].add(()); m = n+1;
    for (i, j, k) in [(x, y, z) for x in range(m) for y in range(m) for z in range(m)]:
        for (l, p) in [(x, y) for x in range(1, i+1) for y in A[i-x][j][k]]:
            A[i][j][k].add(tuple(sorted(list(p)+[(1, l)])))
        for (l, p) in [(x, y) for x in range(1, j+1) for y in A[i][j-x][k]]:
            A[i][j][k].add(tuple(sorted(list(p)+[(1, l)])))
        for (l, p) in [(x, y) for x in range(1, k+1) for y in A[i][j][k-x]]:
            A[i][j][k].add(tuple(sorted(list(p)+[(1, l)])))
        for (l, p) in [(x, y) for x in range(1, min(i, j)+1) for y in A[i-x][j-x][k]]:
            if (i==j): A[i][j][k].add(tuple(sorted(list(p)+[tuple(sorted((2, l)))])))
        for (l, p) in [(x, y) for x in range(1, min(j, k)+1) for y in A[i][j-x][k-x]]:
            if (j==k): A[i][j][k].add(tuple(sorted(list(p)+[tuple(sorted((2, l)))])))
        for (l, p) in [(x, y) for x in range(1, min(i, j, k)+1) for y in A[i-x][j-x][k-x]]:
            if (i==j==k): A[i][j][k].add(tuple(sorted(list(p)+[tuple(sorted((3, l)))])))
    return len(A[n][n][n])  # Robin Visser, May 01 2025

Crossrefs

Third column of A360629.

Cf. A000041 (1 x n rectangle), A360631 (2 x n rectangle).

Sequence in context: A331017 A207646 A071563 * A395156 A139116 A373081

Adjacent sequences: A360629 A360630 A360631 * A360633 A360634 A360635

Keyword

nonn,more

Author

Pontus von Brömssen, Feb 14 2023

Extensions

a(17)-a(20) from Robin Visser, May 04 2025

a(0) = 1 prepended by Robin Visser, May 05 2025

Status

approved