%I #31 Oct 27 2023 19:49:16

%S 1,2,2,3,10,3,5,40,40,5,8,172,369,172,8,13,728,3755,3755,728,13,21,

%T 3096,37320,92801,37320,3096,21,34,13152,373177,2226936,2226936,

%U 373177,13152,34,55,55888,3725843,53841725,128171936,53841725,3725843,55888,55,89

%N T(n,k) = number of ways to reciprocally link elements of an n X k array either to themselves or to exactly one king-move neighbor.

%C Table starts

%C ...1........2............3.................5.....................8

%C ...2.......10...........40...............172...................728

%C ...3.......40..........369..............3755.................37320

%C ...5......172.........3755.............92801...............2226936

%C ...8......728........37320...........2226936.............128171936

%C ..13.....3096.......373177..........53841725............7444342896

%C ..21....13152......3725843........1299348473..........431408410784

%C ..34....55888.....37213728.......31371388772........25014514225856

%C ..55...237472....371654153......757341382671......1450226501771584

%C ..89..1009056...3711809483....18283618480037.....84080327982982848

%C .144..4287616..37070598992...441397115736816...4874715696405194752

%C .233.18218688.370232236753.10656083384666537.282621433306639435392

%H Alois P. Heinz, <a href="/A220644/b220644.txt">Table of n, a(n) for n = 1..528 (antidiagonals 1..32)</a> (terms n = 1..180 from R. H. Hardin)

%e Some solutions for n=3 k=4 0=self 1=nw 2=n 3=ne 4=w 6=e 7=sw 8=s 9=se (reciprocal directions total 10)

%e ..0..6..4..8....6..4..0..0....8..0..0..0....9..6..4..8....6..4..0..0

%e ..0..7..7..2....8..0..9..7....2..8..8..0....8..1..9..2....0..0..8..8

%e ..3..3..6..4....2..0..3..1....0..2..2..0....2..6..4..1....0..0..2..2

%Y Columns k=1-10 give: A000045(n+1), A052978, A220639, A220640, A220641, A220642, A220643, A243314, A243315, A243316.

%Y Main diagonal is A220638.

%Y Cf. A239264.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 17 2012