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Number of (n+1) X 2 0..3 arrays with every 2 X 2 subblock having three or four distinct values, and new values 0..3 introduced in row major order.
1

%I #8 Jul 15 2018 12:03:38

%S 7,78,864,9576,106128,1176192,13035456,144468864,1601114112,

%T 17744767488,196661044224,2179547652096,24155409051648,

%U 267708662341632,2966951532036096,32882019268460544,364423611069038592

%N Number of (n+1) X 2 0..3 arrays with every 2 X 2 subblock having three or four distinct values, and new values 0..3 introduced in row major order.

%C Column 1 of A210413.

%H R. H. Hardin, <a href="/A210406/b210406.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 10*a(n-1) + 12*a(n-2).

%F Conjectures from _Colin Barker_, Jul 15 2018: (Start)

%F G.f.: x*(7 + 8*x) / (1 - 10*x - 12*x^2).

%F a(n) = ((5-sqrt(37))^n*(-11+2*sqrt(37)) + (5+sqrt(37))^n*(11+2*sqrt(37))) / (6*sqrt(37)).

%F (End)

%e Some solutions for n=4:

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

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

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

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

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

%Y Cf. A210413.

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 21 2012