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The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by two tiles that are each fixed under both of these reflections.
6

%I #24 Jul 06 2024 10:19:52

%S 2,9,168,16960,8407040,17180983296,140737630961664,

%T 4611686053860868096,604462909825456529211392,

%U 316912650057075646247661993984,664613997892457973921852429862699008,5575186299632655785536225887234636434636800,187072209578355573530072906199130068813267662274560

%N The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by two tiles that are each fixed under both of these reflections.

%H Peter Kagey, <a href="/A367526/a367526_2.pdf">Illustration of a(2)=9</a>

%H Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv: 2311.13072 [math.CO], 2023. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kagey/kagey6.html">J. Int. Seq.</a>, (2024) Vol. 27, Art. No. 24.6.1, pp. A-6, A-9.

%F a(2m-1) = 2^(2m^2 - 4m - 1)(4^m + 4^m^2 + 8^m).

%F a(2m) = 4^(m^2 - 1)(1 + 2^(1 + m) + 4^m^2).

%t Table[{2^(2 m^2 - 4 m - 1) (4^m + 4^m^2 + 8^m), 4^(m^2 - 1) (1 + 2^(1 + m) + 4^m^2)}, {m, 1, 5}] // Flatten

%Y Cf. A054247, A367526, A367527, A367528, A367529.

%K nonn

%O 1,1

%A _Peter Kagey_, Dec 10 2023