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A223599 T(n,k)=Petersen graph (8,2) coloring a rectangular array: number of nXk 0..15 arrays where 0..15 label nodes of a graph with edges 0,1 0,8 8,14 8,10 1,2 1,9 9,15 9,11 2,3 2,10 10,12 3,4 3,11 11,13 4,5 4,12 12,14 5,6 5,13 13,15 6,7 6,14 7,0 7,15 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph.

%I A223599 #6 Jul 23 2025 04:17:04
%S A223599 16,48,256,144,256,4096,432,1504,1376,65536,1296,6736,16192,7424,
%T A223599 1048576,3888,32768,122608,176224,40160,16777216,11664,156592,1124064,
%U A223599 2372080,1931968,217600,268435456,34992,755200,9902320,43725920,47659632
%N A223599 T(n,k)=Petersen graph (8,2) coloring a rectangular array: number of nXk 0..15 arrays where 0..15 label nodes of a graph with edges 0,1 0,8 8,14 8,10 1,2 1,9 9,15 9,11 2,3 2,10 10,12 3,4 3,11 11,13 4,5 4,12 12,14 5,6 5,13 13,15 6,7 6,14 7,0 7,15 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph.
%C A223599 Table starts
%C A223599 ............16........48..........144.............432...............1296
%C A223599 ...........256.......256.........1504............6736..............32768
%C A223599 ..........4096......1376........16192..........122608............1124064
%C A223599 .........65536......7424.......176224.........2372080...........43725920
%C A223599 .......1048576.....40160......1931968........47659632.........1807461152
%C A223599 ......16777216....217600.....21308000.......982848688........77164934624
%C A223599 .....268435456...1180256....236213312.....20631729648......3355919411936
%C A223599 ....4294967296...6405888...2629972704....438231627440....147579242411936
%C A223599 ...68719476736..34782688..29389265856...9379905920496...6534353238114336
%C A223599 .1099511627776.188912640.329426847840.201754894742320.290550417324168160
%H A223599 R. H. Hardin, <a href="/A223599/b223599.txt">Table of n, a(n) for n = 1..161</a>
%F A223599 Empirical for column k:
%F A223599 k=1: a(n) = 16*a(n-1)
%F A223599 k=2: a(n) = 8*a(n-1) -11*a(n-2) -16*a(n-3)
%F A223599 k=3: a(n) = 23*a(n-1) -153*a(n-2) +217*a(n-3) +258*a(n-4) -456*a(n-5) -104*a(n-6) +192*a(n-7)
%F A223599 k=4: [order 9]
%F A223599 k=5: [order 29]
%F A223599 k=6: [order 55]
%F A223599 Empirical for row n:
%F A223599 n=1: a(n) = 3*a(n-1)
%F A223599 n=2: a(n) = 6*a(n-1) +3*a(n-2) -42*a(n-3) -8*a(n-4) +48*a(n-5) for n>6
%F A223599 n=3: [order 11] for n>12
%F A223599 n=4: [order 28] for n>29
%F A223599 n=5: [order 74] for n>75
%e A223599 Some solutions for n=3 k=4
%e A223599 .14..6..5.13...13.15..9.15...12..4.12.10....6..5.13.15....8.14..8.10
%e A223599 ..7..6..5..6...13.15..9..1...12..4.12..4....6..5.13..5....8.14..8.14
%e A223599 ..5..6.14..6....9.15..9.11....5..4.12.14...13..5..6..5....6.14..6.14
%Y A223599 Column 1 is A001025
%Y A223599 Column 2 is A223434
%Y A223599 Row 1 is A188825(n+1)
%K A223599 nonn,tabl
%O A223599 1,1
%A A223599 _R. H. Hardin_ Mar 23 2013