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

Author

Pontus von Brömssen, Apr 15 2023

Keywords

• nonn

Crossrefs

Third column of A362258.

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

Programs

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))

Formula

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