[seqfan] Coloring the spaces

[Ron Hardin]

Instead of coloring an nXk array with (say) 0..2, one can color the nXk 2X2 subblocks of an (n+1)X(k+1) 0..2 array with some function of their four corner points.

Order the corner points to get minimum, lower median, upper median and maximum.

Among the additive functions based on these, the colorings with the upper median alone come out with particularly simple recurrences (but growing offsets)


/tmp/ead
T(n,k)=Number of (n+1)X(k+1) 0..2 arrays colored with the upper median of every 2X2 subblock

Table starts
......81....294...905..2584..7353.20800.58896.167240.474948.1344000.3811120
.....294....722..1372..2660..5034..9752.18468..35794..67924..131474..251786
.....905...1372..1494..1780..2478..3504..4884...6888...9816...13920...19554
....2584...2660..1780..2356..3496..5036..7320..11908..17592...25318...36632
....7353...5034..2478..3496..5358..6728..9260..14016..20570...26912...36512
...20800...9752..3504..5036..6728.10420.14440..23148..33704...52348...72200
...58896..18468..4884..7320..9260.14440.21288..29328..41748...57760...83202
..167240..35794..6888.11908.14016.23148.29328..49860..70224..108252..146640
..474948..67924..9816.17592.20570.33704.41748..70224.114312..134816..190180
.1344000.131474.13920.25318.26912.52348.57760.108252.134816..228100..288800

Some.solutions.for.n=4.k=4..
..1..0..0..2..2....2..1..0..1..2....0..0..2..0..0....2..0..0..1..2..
..2..1..0..1..1....0..0..0..2..1....1..0..1..2..0....0..1..0..2..0..
..1..2..0..0..0....0..1..2..1..0....2..0..0..0..2....0..0..2..1..0..
..0..0..2..1..0....0..2..0..0..0....0..2..1..0..1....1..2..0..0..0..
..0..1..1..2..1....2..1..0..1..2....1..1..2..0..0....2..1..0..2..2..

Empirical for column k:
k=1: a(n)=8*a(n-3)+36*a(n-4)+48*a(n-6) for n>9
k=2: a(n)=3*a(n-3)+10*a(n-4)-12*a(n-7)-6*a(n-8) for n>12
k=3: a(n)=4*a(n-4) for n>13
k=4: a(n)=5*a(n-4) for n>14
k=5: a(n)=4*a(n-4) for n>13
k=6: a(n)=5*a(n-4) for n>14
k=7: a(n)=4*a(n-4) for n>15

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