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T(n,k)=Number of nXk 0..2 arrays with no element equal to a strict majority of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.
11

%I #4 Dec 15 2016 11:04:47

%S 0,0,0,2,4,0,4,36,40,0,14,304,944,352,0,40,2212,20776,23072,3008,0,

%T 120,15428,406200,1356120,547168,25280,0,352,103648,7630156,72177144,

%U 86246944,12701248,209792,0,1032,680052,138602548,3684310576,12490527012

%N T(n,k)=Number of nXk 0..2 arrays with no element equal to a strict majority of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

%C Table starts

%C .0.......0..........2..............4................14....................40

%C .0.......4.........36............304..............2212.................15428

%C .0......40........944..........20776............406200...............7630156

%C .0.....352......23072........1356120..........72177144............3684310576

%C .0....3008.....547168.......86246944.......12490527012.........1732429706176

%C .0...25280...12701248.....5385546376.....2121871518232.......800037452999320

%C .0..209792..290067328...331573929104...355347019237332....364333887872124232

%C .0.1723392.6540226304.20185283466808.58835479020749472.164074884296083732404

%H R. H. Hardin, <a href="/A279580/b279580.txt">Table of n, a(n) for n = 1..110</a>

%F Empirical for column k:

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

%F k=2: a(n) = 16*a(n-1) -72*a(n-2) +64*a(n-3) -16*a(n-4)

%F k=3: [order 6] for n>7

%F k=4: [order 16] for n>17

%F k=5: [order 28] for n>29

%F Empirical for row n:

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

%F n=2: [order 8]

%F n=3: [order 34] for n>35

%e Some solutions for n=3 k=4

%e ..0..0..1..0. .0..1..1..1. .0..1..1..2. .0..1..2..1. .0..1..2..1

%e ..1..2..1..2. .2..0..0..0. .1..0..0..2. .0..2..2..2. .1..0..0..2

%e ..0..1..0..2. .0..2..2..0. .2..0..2..1. .1..2..1..0. .0..2..2..1

%Y Row 1 is A279322.

%K nonn,tabl

%O 1,4

%A _R. H. Hardin_, Dec 15 2016