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T(n,k)=Number of nXk 0..1 arrays with each 1 horizontally, vertically or antidiagonally adjacent to 1 or 3 neighboring 1s.
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%I #4 Dec 11 2017 14:03:12

%S 1,2,2,3,6,3,4,11,11,4,6,28,36,28,6,9,63,115,115,63,9,13,139,387,597,

%T 387,139,13,19,328,1280,2969,2969,1280,328,19,28,735,4237,14502,21769,

%U 14502,4237,735,28,41,1684,14026,72069,159732,159732,72069,14026,1684,41,60

%N T(n,k)=Number of nXk 0..1 arrays with each 1 horizontally, vertically or antidiagonally adjacent to 1 or 3 neighboring 1s.

%C Table starts

%C ..1....2.....3.......4........6..........9..........13............19

%C ..2....6....11......28.......63........139.........328...........735

%C ..3...11....36.....115......387.......1280........4237.........14026

%C ..4...28...115.....597.....2969......14502.......72069........355733

%C ..6...63...387....2969....21769.....159732.....1181174.......8690359

%C ..9..139..1280...14502...159732....1740403....19142885.....209776696

%C .13..328..4237...72069..1181174...19142885...313843472....5120370505

%C .19..735.14026..355733..8690359..209776696..5120370505..124433749976

%C .28.1684.46454.1758440.64100854.2301194495.83668287300.3028311394674

%H R. H. Hardin, <a href="/A296396/b296396.txt">Table of n, a(n) for n = 1..312</a>

%F Empirical for column k:

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

%F k=2: a(n) = a(n-1) +2*a(n-2) +2*a(n-3) -a(n-4) +4*a(n-5) -a(n-6) -4*a(n-7) +a(n-9)

%F k=3: [order 14]

%F k=4: [order 49]

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A000930(n+1).

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 11 2017