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%I #5 Apr 16 2023 08:38:09
%S 1,1,1,1,2,4,8,12,22,40,73,146,292,560,1120,2532,5040,10080,22176,
%T 44352,88704,192272,384384,768768,1647360,3294720,6589440,14003120,
%U 28006240,56012480,126028080,266053680,532107360,1182438400,2483130720,4966261440,10925775168
%N Maximum number of ways in which a set of integer-sided squares can tile an n X 3 rectangle, up to rotations and reflections.
%F a(n) >= A362144(n)/4.
%o (Python)
%o from math import comb
%o def F(i,j,k):
%o # total number of tilings using i, j, and 2*j+3*k squares of side lengths 3, 2, and 1, respectively
%o return comb(i+j+k,i)*comb(j+k,j)*2**j
%o def F0(i,j,k):
%o # number of inequivalent tilings
%o x = F(i,j,k)
%o if j == 0: x += comb(i+k,i) # horizontal line of symmetry
%o if i%2+j%2+k%2 <= 1: x += 2*F(i//2,j//2,k//2) # vertical line of symmetry or rotational symmetry
%o return x//4
%o def A362261(n):
%o 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))
%Y Third column of A362258.
%Y Cf. A359019, A361225 (rectangular pieces), A362144.
%K nonn
%O 0,5
%A _Pontus von Brömssen_, Apr 15 2023
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