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T(n,k) = Number of n X k 0..1 arrays with every element equal to 2, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero.
7

%I #6 Apr 22 2021 14:53:04

%S 0,0,0,0,1,0,0,3,3,0,0,6,8,6,0,0,17,36,36,17,0,0,41,173,263,173,41,0,

%T 0,104,858,2537,2537,858,104,0,0,261,4258,22718,46286,22718,4258,261,

%U 0,0,655,21386,214683,816886,816886,214683,21386,655,0,0,1646,107465,2024559

%N T(n,k) = Number of n X k 0..1 arrays with every element equal to 2, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero.

%C Table starts

%C .0...0.....0.......0.........0...........0.............0...............0

%C .0...1.....3.......6........17..........41...........104.............261

%C .0...3.....8......36.......173.........858..........4258...........21386

%C .0...6....36.....263......2537.......22718........214683.........2024559

%C .0..17...173....2537.....46286......816886......14783424.......267652693

%C .0..41...858...22718....816886....27685946.....967671172.....33782479865

%C .0.104..4258..214683..14783424...967671172...65181402152...4383565657986

%C .0.261.21386.2024559.267652693.33782479865.4383565657986.567774280392040

%H R. H. Hardin, <a href="/A299334/b299334.txt">Table of n, a(n) for n = 1..180</a>

%F Empirical for column k:

%F k=1: a(n) = a(n-1);

%F k=2: a(n) = a(n-1) +3*a(n-2) +2*a(n-3);

%F k=3: [order 12] for n > 14;

%F k=4: [order 27] for n > 28;

%F k=5: [order 76] for n > 79.

%e Some solutions for n=5, k=4

%e ..0..0..1..1. .0..0..1..1. .0..0..0..0. .0..0..1..1. .0..0..0..0

%e ..1..0..1..1. .1..0..1..1. .0..1..1..0. .0..0..1..0. .0..0..0..0

%e ..1..1..0..0. .1..1..1..1. .0..1..0..1. .1..0..0..0. .1..1..1..0

%e ..1..0..1..0. .1..0..0..1. .1..0..1..1. .1..1..0..0. .0..1..0..1

%e ..0..0..1..1. .0..0..1..1. .1..1..1..1. .1..0..0..0. .0..0..1..1

%Y Column 2 is A297972.

%K nonn,tabl

%O 1,8

%A _R. H. Hardin_, Feb 07 2018