A229694 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.
0, 0, 0, 1, 3, 0, 3, 43, 40, 0, 12, 245, 626, 336, 0, 40, 1171, 5077, 6732, 2304, 0, 120, 5077, 35825, 80757, 62856, 14080, 0, 336, 20691, 230383, 848937, 1125333, 539568, 79872, 0, 896, 80757, 1400413, 8186713, 17724789, 14461173, 4377888, 430080, 0
Offset: 1
Examples
Some solutions for n=3, k=4: 0 1 2 1 0 1 0 2 0 1 0 2 0 0 1 2 0 1 0 2 0 1 0 2 0 2 0 2 0 2 1 0 1 0 0 1 0 2 1 1 1 2 1 1 2 1 2 0 2 2 1 2 2 1 2 0 1 0 2 2 Table starts .0......0........1..........3...........12............40.............120 .0......3.......43........245.........1171..........5077...........20691 .0.....40......626.......5077........35825........230383.........1400413 .0....336.....6732......80757.......848937.......8186713........75035643 .0...2304....62856....1125333.....17724789.....258006388......3583403667 .0..14080...539568...14461173....342532665....7551515197....159377253183 .0..79872..4377888..175867605...6279934941..210095323918...6749642728251 .0.430080.34105536.2054728053.110801828529.5632122625852.275739382892979
Links
- R. H. Hardin, Table of n, a(n) for n = 1..286
Formula
Empirical for column k:
k=1: a(n) = a(n-1).
k=2: a(n) = 12*a(n-1) - 48*a(n-2) + 64*a(n-3).
k=3: a(n) = 18*a(n-1) - 108*a(n-2) + 216*a(n-3) for n > 4.
k=4: a(n) = 27*a(n-1) - 243*a(n-2) + 729*a(n-3) for n > 4.
k=5: [order 6] for n > 7.
k=6: [order 9] for n > 11.
k=7: [order 12] for n > 14.
Empirical for row n:
n=1: a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n > 6.
n=2: a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) for n > 6.
n=3: a(n) = 15*a(n-1) - 81*a(n-2) + 185*a(n-3) - 162*a(n-4) + 60*a(n-5) - 8*a(n-6) for n > 10.
n=4: [order 9] for n > 17.
n=5: [order 21] for n > 27.
n=6: [order 29] for n > 39.
n=7: [order 86] for n > 94.