OFFSET
1,1
COMMENTS
From Nicolas Bělohoubek, Apr 16 2026: (Start)
The number of ways to tile a 3n X 3n square using I-trominoes can be bounded from above by replacing each I-tromino with 2 arrows and circle. The center of an I-tromino is marked as a circle, and its 2 remaining squares are marked as arrows pointing toward the center.
If we count all possible arrangements of arrows and circles in a 3n X 3n grid, we get 5^((3n)^2). However, this is far too large.
Let us focus on which local configurations in a 3 X 3 square are allowed. We have 5^9 possible configurations, but we restrict ourselves to those satisfying the following 4 conditions:
1. No arrow points to another arrow.
2. If a circle is in a corner of the 3 X 3 square, exactly 1 arrow must point to it.
3. If a circle is not in a corner, exactly 2 arrows must point to it, and these arrows must come from opposite directions.
4. An arrow is allowed to point outside the 3 X 3 square.
The number of such valid configurations is only 162. Therefore, the number of tilings is at most 162^(n^2). (End)
LINKS
J. Van Craen, The residual entropy of rectilinear trimers on the square lattice at close packing, J. Chem. Phys. 63 (1975) 2591-2596.
FORMULA
From Nicolas Bělohoubek, Apr 16 2026: (Start)
The four color theorem implies, that a(n) <= 4^((3n)^2) = 262144^(n^2).
a(n) <= 162^(n^2). (See comments.)
Conjecture: a(n) ~ alpha * beta^n * gamma^(n^2), where 0.933 < alpha < 0.979, 0.4619 < beta < 0.4696, 4.1691 < gamma < 4.1739. Notice similarity with A004003. (End)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Steven Finch, May 20 2008
EXTENSIONS
Name corrected and a(6)-a(7) from Andrew Howroyd, Feb 16 2022
STATUS
approved