A362261 Maximum number of ways in which a set of integer-sided squares can tile an n X 3 rectangle, up to rotations and reflections.

1, 1, 1, 1, 2, 4, 8, 12, 22, 40, 73, 146, 292, 560, 1120, 2532, 5040, 10080, 22176, 44352, 88704, 192272, 384384, 768768, 1647360, 3294720, 6589440, 14003120, 28006240, 56012480, 126028080, 266053680, 532107360, 1182438400, 2483130720, 4966261440, 10925775168

Offset

0,5

Links

Table of n, a(n) for n=0..36.

Formula

a(n) >= A362144(n)/4.

Prog

(Python)
from math import comb
def F(i, j, k):
    # total number of tilings using i, j, and 2*j+3*k squares of side lengths 3, 2, and 1, respectively
    return comb(i+j+k, i)*comb(j+k, j)*2**j
def F0(i, j, k):
    # number of inequivalent tilings
    x = F(i, j, k)
    if j == 0: x += comb(i+k, i) # horizontal line of symmetry
    if i%2+j%2+k%2 <= 1: x += 2*F(i//2, j//2, k//2) # vertical line of symmetry or rotational symmetry
    return x//4
def A362261(n):
    return max(F0(i, j, n-3*i-2*j) for i in range(n//3+1) for j in range((n-3*i)//2+1))

Crossrefs

Third column of A362258.

Cf. A359019, A361225 (rectangular pieces), A362144.

Sequence in context: A365076 A327480 A330022 * A032473 A084422 A175841

Adjacent sequences: A362258 A362259 A362260 * A362262 A362263 A362264

Keyword

nonn

Author

Pontus von Brömssen, Apr 15 2023

Status

approved