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%I #6 Sep 03 2022 20:38:27
%S 3,6,9,12,24,27,24,72,96,81,48,216,432,384,243,96,648,1944,2592,1536,
%T 729,192,1944,8856,17496,15552,6144,2187,384,5832,40392,121176,157464,
%U 93312,24576,6561,768,17496,184248,842616,1658232,1417176,559872,98304,19683
%N T(n,k) = Number of n X k 0..2 arrays with no element x(i,j) adjacent to value 2-x(i,j) horizontally or antidiagonally.
%C Table starts
%C .....3.......6........12.........24...........48.............96
%C .....9......24........72........216..........648...........1944
%C ....27......96.......432.......1944.........8856..........40392
%C ....81.....384......2592......17496.......121176.........842616
%C ...243....1536.....15552.....157464......1658232.......17587584
%C ...729....6144.....93312....1417176.....22692312......367125912
%C ..2187...24576....559872...12754584....310536504.....7663517136
%C ..6561...98304...3359232..114791256...4249585944...159971190624
%C .19683..393216..20155392.1033121304..58154132088..3339300422232
%C .59049.1572864.120932352.9298091736.795819434328.69705848287656
%H R. H. Hardin, <a href="/A233155/b233155.txt">Table of n, a(n) for n = 1..390</a>
%F Empirical for column k:
%F k=1: a(n) = 3*a(n-1).
%F k=2: a(n) = 4*a(n-1).
%F k=3: a(n) = 6*a(n-1).
%F k=4: a(n) = 9*a(n-1).
%F k=5: a(n) = 15*a(n-1) -18*a(n-2).
%F k=6: a(n) = 25*a(n-1) -90*a(n-2) +81*a(n-3).
%F k=7: a(n) = 42*a(n-1) -351*a(n-2) +972*a(n-3) -810*a(n-4).
%F Empirical for row n:
%F n=1: a(n) = 2*a(n-1).
%F n=2: a(n) = 3*a(n-1) for n>2.
%F n=3: a(n) = 5*a(n-1) -2*a(n-2) for n>4.
%F n=4: a(n) = 9*a(n-1) -15*a(n-2) +6*a(n-3) for n>7.
%F n=5: [order 7] for n>11.
%F n=6: [order 9] for n>15.
%F n=7: [order 27] for n>33.
%e Some solutions for n=4, k=4
%e ..1..2..2..1....1..2..2..1....0..0..0..0....1..2..1..0....2..1..0..1
%e ..2..1..2..2....2..1..0..1....1..0..1..2....1..0..0..0....0..1..0..1
%e ..2..1..2..1....2..1..0..1....1..0..1..0....0..0..1..0....2..1..2..1
%e ..0..1..2..2....0..1..0..0....1..0..0..1....0..0..0..0....2..1..0..0
%Y Column 1 is A000244.
%Y Column 2 is A002023(n-1).
%Y Column 3 is 2*A000400.
%Y Column 4 is 3*A055275.
%Y Row 1 is A003945.
%Y Row 2 is A005051(n-1) for n>1.
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Dec 05 2013
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