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A362261
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Maximum number of ways in which a set of integer-sided squares can tile an n X 3 rectangle, up to rotations and reflections.
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2
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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
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OFFSET
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0,5
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LINKS
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FORMULA
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PROG
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(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
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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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