A198405 - OEIS

%I #9 Mar 02 2018 06:27:10

%S 2,17,377,11473,375273,12456897,414711897,13814539697,460231956937,

%T 15333001667233,510833776539193,17018936996199057,567002973887727017,

%U 18890274083549781377,629348476275500726297,20967377362990867086193

%N Number of 2n X 2 0..4 arrays with values 0..4 introduced in row major order and each element equal to exactly one horizontal and vertical neighbor.

%C Column 1 of A198408.

%H R. H. Hardin, <a href="/A198405/b198405.txt">Table of n, a(n) for n = 1..200</a>

%F Empirical: a(n) = 51*a(n-1) - 691*a(n-2) + 3601*a(n-3) - 7056*a(n-4) + 4096*a(n-5).

%F Conjectures from _Colin Barker_, Mar 02 2018: (Start)

%F G.f.: x*(2 - 85*x + 892*x^2 - 3209*x^3 + 3552*x^4) / ((1 - x)*(1 - 9*x + 16*x^2)*(1 - 41*x + 256*x^2)).

%F a(n) = (2^(-8-n)*(-1752*(9-sqrt(17))^n*(-51+5*sqrt(17)) + 1752*(9+sqrt(17))^n*(51+5*sqrt(17)) + 17*(73*2^(9+n) + (365-17*sqrt(73))*(41-3*sqrt(73))^n + (41+3*sqrt(73))^n*(365+17*sqrt(73))))) / 3723.

%F (End)

%e Some solutions for n=3:

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

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

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

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

%e ..4..0....3..2....1..0....3..0....4..4....2..0....4..4....1..3....0..0....4..4

%e ..4..0....3..2....1..0....3..0....2..2....1..1....2..2....1..3....1..1....3..3

%Y Cf. A198408.

%K nonn

%O 1,1

%A _R. H. Hardin_, Oct 24 2011