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A232955 T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally, diagonally or antidiagonally, and top left element zero.

%I A232955 #6 Jul 23 2025 07:45:58
%S A232955 1,3,4,9,21,16,27,129,147,64,81,771,1881,1029,256,243,4629,22971,
%T A232955 27441,7203,1024,729,27771,283131,685251,400329,50421,4096,2187,
%U A232955 166629,3484893,17429409,20442651,5840289,352947,16384,6561,999771,42904365,442227825
%N A232955 T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally, diagonally or antidiagonally, and top left element zero.
%C A232955 Table starts
%C A232955 ......1.........3............9..............27.................81
%C A232955 ......4........21..........129.............771...............4629
%C A232955 .....16.......147.........1881...........22971.............283131
%C A232955 .....64......1029........27441..........685251...........17429409
%C A232955 ....256......7203.......400329........20442651.........1074244299
%C A232955 ...1024.....50421......5840289.......609853251........66226131273
%C A232955 ...4096....352947.....85202361.....18193384251......4082986991091
%C A232955 ..16384...2470629...1242993681....542752261251....251727862281441
%C A232955 ..65536..17294403..18133691049..16191600916251..15519780149309307
%C A232955 .262144.121060821.264547403649.483034266181251.956841601733733945
%H A232955 R. H. Hardin, <a href="/A232955/b232955.txt">Table of n, a(n) for n = 1..199</a>
%F A232955 Empirical for column k:
%F A232955 k=1: a(n) = 4*a(n-1)
%F A232955 k=2: a(n) = 7*a(n-1)
%F A232955 k=3: a(n) = 15*a(n-1) -6*a(n-2)
%F A232955 k=4: a(n) = 31*a(n-1) -35*a(n-2) +5*a(n-3)
%F A232955 k=5: [order 10]
%F A232955 k=6: [order 21]
%F A232955 Empirical for row n:
%F A232955 n=1: a(n) = 3*a(n-1)
%F A232955 n=2: a(n) = 5*a(n-1) +6*a(n-2)
%F A232955 n=3: a(n) = 12*a(n-1) +7*a(n-2) -40*a(n-3) +12*a(n-4)
%F A232955 n=4: [order 10]
%F A232955 n=5: [order 26] for n>27
%F A232955 n=6: [order 86] for n>87
%e A232955 Some solutions for n=3 k=4
%e A232955 ..0..2..3..2....0..1..3..3....0..1..0..2....0..1..0..0....0..2..0..2
%e A232955 ..2..2..3..3....0..1..1..1....3..2..0..1....0..2..0..0....2..0..2..2
%e A232955 ..0..2..3..3....0..1..0..0....2..2..3..2....0..2..0..2....1..0..2..0
%Y A232955 Column 1 is A000302(n-1)
%Y A232955 Column 2 is A169634(n-1)
%Y A232955 Row 1 is A000244(n-1)
%K A232955 nonn,tabl
%O A232955 1,2
%A A232955 _R. H. Hardin_, Dec 02 2013