A368222 - OEIS

A368222

Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by an asymmetric tile.

%I #13 Oct 19 2024 21:54:51

%S 1,2,3,4,10,4,8,36,32,10,16,136,256,136,16,32,528,2048,2080,512,36,64,

%T 2080,16384,32896,16384,2080,64,128,8256,131072,524800,524288,131328,

%U 8192,136,256,32896,1048576,8390656,16777216,8390656,1048576,32896,256

%N Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by an asymmetric tile.

%H Peter Kagey, <a href="/A368222/a368222.pdf">Illustration of T(3,2)=32</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 4 8 16 32

%e 2 | 3 10 36 136 528 2080

%e 3 | 4 32 256 2048 16384 131072

%e 4 | 10 136 2080 32896 524800 8390656

%e 5 | 16 512 16384 524288 16777216 536870912

%e 6 | 36 2080 131328 8390656 536887296 34359869440

%t A368222[n_, m_] := 2^(n*m/2 - 1) (2^(n*m/2) + Boole[EvenQ[n]])

%Y Cf. A368220, A368221, A368224, A117401.

%K nonn,tabl

%O 1,2

%A _Peter Kagey_, Dec 18 2023