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A223305 T(n,k)=4X4X4 triangular graph coloring a rectangular array: number of nXk 0..9 arrays where 0..9 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 3,6 3,7 4,7 6,7 4,8 5,8 7,8 5,9 8,9 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

%I A223305 #6 Jul 23 2025 03:58:53
%S A223305 10,36,36,144,324,144,576,3132,3132,576,2304,30564,76788,30564,2304,
%T A223305 9216,298620,1882500,1882500,298620,9216,36864,2918052,46589268,
%U A223305 118001928,46589268,2918052,36864,147456,28515132,1148040516,7442199324,7442199324
%N A223305 T(n,k)=4X4X4 triangular graph coloring a rectangular array: number of nXk 0..9 arrays where 0..9 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 3,6 3,7 4,7 6,7 4,8 5,8 7,8 5,9 8,9 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.
%C A223305 Table starts
%C A223305 .....10........36..........144..............576................2304
%C A223305 .....36.......324.........3132............30564..............298620
%C A223305 ....144......3132........76788..........1882500............46589268
%C A223305 ....576.....30564......1882500........118001928..........7442199324
%C A223305 ...2304....298620.....46589268.......7442199324.......1208825525124
%C A223305 ...9216...2918052...1148040516.....470062209228.....195984333263964
%C A223305 ..36864..28515132..28402185108...29706324866280...31982581839418764
%C A223305 .147456.278649828.700525554180.1877636655597516.5198236290762699924
%H A223305 R. H. Hardin, <a href="/A223305/b223305.txt">Table of n, a(n) for n = 1..161</a>
%F A223305 Empirical for column k:
%F A223305 k=1: a(n) = 4*a(n-1) for n>2
%F A223305 k=2: a(n) = 11*a(n-1) -12*a(n-2)
%F A223305 k=3: [order 11] for n>12
%F A223305 k=4: [order 25] for n>26
%e A223305 Some solutions for n=3 k=4
%e A223305 ..5..2..4..2....5..2..5..4....0..2..0..2....0..1..4..7....0..2..4..7
%e A223305 ..2..5..2..0....2..5..4..8....2..1..2..5....2..0..2..4....2..4..1..4
%e A223305 ..0..2..4..1....5..4..2..4....4..2..5..8....1..2..5..8....4..3..4..8
%Y A223305 Column 1 is A002063(n-1) for n>1
%K A223305 nonn,tabl
%O A223305 1,1
%A A223305 _R. H. Hardin_ Mar 19 2013