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T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 2, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero.
7

%I #4 Jul 21 2018 17:34:01

%S 1,2,2,4,8,4,8,23,23,8,16,65,82,65,16,32,192,302,302,192,32,64,569,

%T 1164,1597,1164,569,64,128,1709,4561,8598,8598,4561,1709,128,256,5162,

%U 17966,46955,65099,46955,17966,5162,256,512,15663,70913,258241,507378,507378

%N T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 2, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero.

%C Table starts

%C ...1.....2......4.......8........16.........32...........64............128

%C ...2.....8.....23......65.......192........569.........1709...........5162

%C ...4....23.....82.....302......1164.......4561........17966..........70913

%C ...8....65....302....1597......8598......46955.......258241........1422755

%C ..16...192...1164....8598.....65099.....507378......3982123.......31333222

%C ..32...569...4561...46955....507378....5694906.....64566869......734571604

%C ..64..1709..17966..258241...3982123...64566869...1062867065....17593640188

%C .128..5162..70913.1422755..31333222..734571604..17593640188...424977120539

%C .256.15663.280153.7843425.246848528.8373046468.292154717629.10317078557668

%H R. H. Hardin, <a href="/A317125/b317125.txt">Table of n, a(n) for n = 1..180</a>

%F Empirical for column k:

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

%F k=2: a(n) = 3*a(n-1) +3*a(n-2) -6*a(n-3) -8*a(n-4) for n>5

%F k=3: [order 16]

%F k=4: [order 57] for n>59

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A000079(n-1).

%Y Column 2 is A304304.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Jul 21 2018