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T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally, diagonally or antidiagonally adjacent to 0 or 2 neighboring 1s.
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%I #4 Dec 30 2017 11:09:50

%S 2,3,4,5,8,8,8,25,19,16,13,57,92,48,32,21,156,285,390,120,64,34,384,

%T 1143,1677,1652,299,128,55,1009,3933,10579,9774,6888,747,256,89,2545,

%U 14817,52714,97005,55713,28971,1865,512,144,6580,52868,303202,693884,862112

%N T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally, diagonally or antidiagonally adjacent to 0 or 2 neighboring 1s.

%C Table starts

%C ...2....3......5........8........13..........21...........34.............55

%C ...4....8.....25.......57.......156.........384.........1009...........2545

%C ...8...19.....92......285......1143........3933........14817..........52868

%C ..16...48....390.....1677.....10579.......52714.......303202........1595942

%C ..32..120...1652.....9774.....97005......693884......6088511.......46944276

%C ..64..299...6888....55713....862112.....8800153....116817805.....1310421193

%C .128..747..28971...321030...7800563...113851142...2300975096....37653648610

%C .256.1865.121854..1848151..70510494..1469935404..45221328552..1078420953679

%C .512.4656.512165.10625522.636242985.18934230754.886373249313.30786030956559

%H R. H. Hardin, <a href="/A297457/b297457.txt">Table of n, a(n) for n = 1..312</a>

%F Empirical for column k:

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

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

%F k=3: a(n) = 5*a(n-1) -3*a(n-2) +4*a(n-3) -25*a(n-4) +10*a(n-5) -7*a(n-6) +12*a(n-7)

%F k=4: [order 11]

%F k=5: [order 27]

%F k=6: [order 55]

%F Empirical for row n:

%F n=1: a(n) = a(n-1) +a(n-2)

%F n=2: a(n) = a(n-1) +3*a(n-2) +a(n-3) +4*a(n-4)

%F n=3: a(n) = 3*a(n-1) +4*a(n-2) -7*a(n-3) +8*a(n-4) -19*a(n-5) +8*a(n-6) +2*a(n-8)

%F n=4: [order 15]

%F n=5: [order 33]

%F n=6: [order 72]

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A000079.

%Y Column 2 is A295045.

%Y Row 1 is A000045(n+2).

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Dec 30 2017