login
The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is not fixed under any of the symmetries of the square.
7

%I #21 Jul 06 2024 10:19:49

%S 1,538,16777216,35184378381312,4722366482869645213696,

%T 40564819207303347603293977182208,

%U 22300745198530623141535718272648361505980416,784637716923335095479473677930668862955643627524327473152,1766847064778384329583297500742918515827483896875618958121606201292619776

%N The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is not fixed under any of the symmetries of the square.

%H Peter Kagey, <a href="/A367525/a367525_1.pdf">Illustration of a(2)=538</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-8.

%F a(2m-1) = 4096^(m^2 - m).

%F a(2m) = 8^(m^2 - 1)*(512^m^2 + 3*8^m^2 + 2).

%t Table[{4096^(m^2 - m), 8^(m^2 - 1) (512^m^2 + 3*8^m^2 + 2)}, {m, 1, 5}] // Flatten

%Y Cf. A054247, A295229, A302484, A367522, A367523, A367524.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 10 2023