A368255 - OEIS

A368255

Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by a tile that is fixed under vertical reflections but not horizontal reflections.

%I #11 Oct 19 2024 21:54:51

%S 1,2,2,3,5,2,6,14,9,4,10,44,50,26,4,20,152,366,298,62,9,36,560,2780,

%T 4244,1692,205,10,72,2144,22028,66184,52740,11272,623,22,136,8384,

%U 175128,1050896,1679368,701124,75486,2171,30

%N Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by a tile that is fixed under vertical reflections but not horizontal reflections.

%H Peter Kagey, <a href="/A368255/a368255.pdf">Illustration of T(2,3)=14</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.

%e Table begins:

%e n\k| 1 2 3 4 5 6

%e ---+---------------------------------------------

%e 1 | 1 2 3 6 10 20

%e 2 | 2 5 14 44 152 560

%e 3 | 2 9 50 366 2780 22028

%e 4 | 4 26 298 4244 66184 1050896

%e 5 | 4 62 1692 52740 1679368 53696936

%e 6 | 9 205 11272 701124 44761184 2863442960

%e 7 | 10 623 75486 9591666 1227208420 157073688884

%t A368255[n_, m_] := 1/(4n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*(2^(n*m/2 - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^((n*m - n)/LCM[d, 2])*2^(n/d)]]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])

%Y Cf. A368218, A368253, A368254, A368256, A368257, A368260.

%K nonn,tabl

%O 1,2

%A _Peter Kagey_, Dec 21 2023