cp's OEIS Frontend

A367522 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under both horizontal and vertical reflection, but not diagonal reflection.

%I A367522 #23 Jul 06 2024 09:57:57
%S A367522 1,4,84,8292,4203520,8590033024,70368815480832,2305843010824323072,
%T A367522 302231454912728264605696,158456325028529097399561355264,
%U A367522 332306998946228986960926214931349504,2787593149816327892693735671512138485071872,93536104789177786765036453099565034406633831137280
%N A367522 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under both horizontal and vertical reflection, but not diagonal reflection.
%C A367522 The a(2) = 4 tilings are
%C A367522   - -   - -   - |       - |
%C A367522   - -,  | -,  - |, and  | -.
%H A367522 Michael De Vlieger, <a href="/A367522/b367522.txt">Table of n, a(n) for n = 1..57</a>
%H A367522 Peter Kagey, <a href="/A367522/a367522.pdf">Illustration of a(3)=84</a>.
%H A367522 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, p. A-6.
%F A367522 a(2m-1) = 4^(m^2 - 2m - 1)*(4^m + 4^m^2 + 8^m).
%F A367522 a(2m)   = 2^(m^2 - 3)*(2 + 3*2^m^2 + 8^m^2).
%t A367522 a[n_] := If[EvenQ[n], 2^(#^2 - 3)*(2 + 3*2^#^2 + 8^#^2) &[n/2], 4^(#^2 - 2 # - 1)*(4^# + 4^#^2 + 8^#) &[(n + 1)/2]]; Array[a, 13] (* _Michael De Vlieger_, Jul 06 2024 *)
%Y A367522 Cf. A054247, A295229, A302484, A367523, A367524, A367525.
%K A367522 nonn
%O A367522 1,2
%A A367522 _Peter Kagey_, Nov 21 2023