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T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-2,1) or (-1,0) and new values introduced in order 0..2.
12

%I #4 Jul 20 2016 18:28:39

%S 1,2,1,5,6,2,14,36,11,4,41,216,61,27,8,122,1296,339,187,66,16,365,

%T 7776,1885,1302,648,162,32,1094,46656,10483,9075,6448,2282,404,64,

%U 3281,279936,58301,63267,64248,32388,8134,1007,128,9842,1679616,324243,441090,640250

%N T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-2,1) or (-1,0) and new values introduced in order 0..2.

%C Table starts

%C ...1....2......5.......14.........41.........122...........365............1094

%C ...1....6.....36......216.......1296........7776.........46656..........279936

%C ...2...11.....61......339.......1885.......10483.........58301..........324243

%C ...4...27....187.....1302.......9075.......63267........441090.........3075255

%C ...8...66....648.....6448......64248......640250.......6380362........63583084

%C ..16..162...2282....32388.....460034.....6534900......92831330......1318716132

%C ..32..404...8134...164645....3334854....67550874....1368326231.....27717164160

%C ..64.1007..29027...844176...24595019...716735348...20887845278....608741311725

%C .128.2512.103456..4319300..181187962..7605814181..319348056936..13409199459965

%C .256.6271.368889.22083548.1332436071.80496567059.4865299894416.294093896351121

%H R. H. Hardin, <a href="/A275228/b275228.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) for n>2

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

%F k=3: [order 9] for n>13

%F k=4: [order 31] for n>35

%F k=5: [order 81] for n>86

%F Empirical for row n:

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

%F n=2: a(n) = 6*a(n-1)

%F n=3: a(n) = 7*a(n-1) -8*a(n-2)

%F n=4: a(n) = 9*a(n-1) -15*a(n-2) +6*a(n-3)

%F n=5: a(n) = 11*a(n-1) -9*a(n-2) -15*a(n-3) +20*a(n-4) -6*a(n-5) for n>6

%F n=6: a(n) = 19*a(n-1) -75*a(n-2) +101*a(n-3) -44*a(n-4) for n>5

%F n=7: a(n) = 20*a(n-1) +57*a(n-2) -1206*a(n-3) +3096*a(n-4) +2306*a(n-5) -16957*a(n-6) +20440*a(n-7) -7755*a(n-8) for n>9

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 1 is A000079(n-2).

%Y Row 1 is A007051(n-1).

%Y Row 2 is A000400(n-1).

%Y Row 4 is A078100.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Jul 20 2016