[seqfan] Single Fault Colorings - equal recurrences?

[Ron Hardin]

The case of a single fault in a 4-coloring of a king-move nXk array leads to small equal recurrences.

Apparently single faults can only happen around the boundary.


/tmp/dkb
T(n,k)=Number of defective 4-colorings of an nXk 0..3 array connected horizontally, vertically, diagonally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..3 order

Table starts
...0....1....2....6...20...70..246..854..2920..9846.32810.108262.354300.1151462
...1....6...26...80..216..544.1312.3072..7040.15872.35328..77824.169984........
...2...26...76..171..362..757.1584.3323..6982.14673.30812..64615...............
...6...80..171..312..568.1064.2064.4120..8384.17256.35728......................
..20..216..362..568..924.1576.2852.5440.10780.21880............................
..70..544..757.1064.1576.2440.4048.7224.13696..................................
.246.1312.1584.2064.2852.4048.6108.9992........................................
.854.3072.3323.4120.5440.7224.9992.............................................

Some.solutions.for.4X4..
..0..1..2..0....0..1..2..3....0..1..0..2....0..1..0..1....0..1..0..1..
..2..1..3..1....2..3..0..1....3..2..3..1....2..3..2..3....2..3..2..3..
..3..0..2..0....0..1..2..3....0..1..0..2....1..0..1..0....1..0..1..0..
..2..1..3..1....2..3..0..2....2..3..1..3....1..2..3..2....0..2..3..2..

Empirical for column k:
k=1: a(n)=8*a(n-1)-22*a(n-2)+24*a(n-3)-9*a(n-4) for n>6
k=2: a(n)=4*a(n-1)-4*a(n-2) for n>3
k=3: a(n)=6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4) for n>5
k=4: a(n)=6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4) for n>7
k=5: a(n)=6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4) for n>7
k=6: a(n)=6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4) for n>7
k=7: a(n)=6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4) for n>7

All.coloring.solutions.for.4X4.WITHOUT.ANY FAULTS..
..0..1..0..1....0..1..2..1....0..1..0..1....0..1..0..1....0..1..0..1..
..2..3..2..3....2..3..0..3....2..3..2..3....2..3..2..3....2..3..2..3..
..1..0..1..0....0..1..2..1....0..1..0..1....0..1..0..1....1..0..1..0..
..3..2..3..2....2..3..0..3....2..3..2..3....3..2..3..2....2..3..2..3..
..
..0..1..2..3....0..1..0..2..
..2..3..0..1....3..2..3..1..
..0..1..2..3....0..1..0..2..
..2..3..0..1....3..2..3..1..


Nothing so simple seems to show up yet in other topologies.

 
rhhardin at mindspring.com
rhhardin at att.net (either)