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Number of defective 4-colorings of an n X 2 0..3 array connected horizontally, vertically, diagonally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..3 order.
1

%I #12 Oct 21 2022 11:10:03

%S 1,6,26,80,216,544,1312,3072,7040,15872,35328,77824,169984,368640,

%T 794624,1703936,3637248,7733248,16384000,34603008,72876032,153092096,

%U 320864256,671088640,1400897536,2919235584,6073352192,12616466432,26172456960

%N Number of defective 4-colorings of an n X 2 0..3 array connected horizontally, vertically, diagonally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..3 order.

%H R. H. Hardin, <a href="/A229572/b229572.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 4*a(n-1) - 4*a(n-2) for n>3.

%F Conjectures from _Colin Barker_, Sep 19 2018: (Start)

%F G.f.: x*(1 + 2*x + 6*x^2) / (1 - 2*x)^2.

%F a(n) = 2^(n-2) * (7*n-8) for n>1. (End)

%F Conjectured e.g.f.: 2 + (3*x + exp(2*x)*(7*x - 4))/2. - _Stefano Spezia_, Oct 21 2022

%e Some solutions for n=3:

%e ..0..1....0..1....0..0....0..1....0..1....0..1....0..1....0..1....0..1....0..1

%e ..2..2....2..0....1..2....2..2....2..1....2..2....2..2....2..3....2..1....2..2

%e ..0..3....1..3....3..0....0..1....3..0....3..1....3..0....2..0....0..3....1..3

%Y Column 2 of A229578.

%K nonn

%O 1,2

%A _R. H. Hardin_, Sep 26 2013