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A199127 Number of n X 2 0..2 arrays with values 0..2 introduced in row major order, the number of instances of each value within one of each other, and no element equal to any horizontal or vertical neighbor.

%I A199127 #20 Oct 13 2021 10:50:01
%S A199127 1,2,2,12,30,30,210,560,560,4200,11550,11550,90090,252252,252252,
%T A199127 2018016,5717712,5717712,46558512,133024320,133024320,1097450640,
%U A199127 3155170590,3155170590,26293088250,75957810500,75957810500,638045608200
%N A199127 Number of n X 2 0..2 arrays with values 0..2 introduced in row major order, the number of instances of each value within one of each other, and no element equal to any horizontal or vertical neighbor.
%C A199127 Column 2 of A199133.
%C A199127 a(n) is the last term in row n of triangle in A286030 (see also formulas below). _Bob Selcoe_, Sep 26 2021
%H A199127 R. H. Hardin, <a href="/A199127/b199127.txt">Table of n, a(n) for n = 1..198</a>
%F A199127 Conjecture: a(3n+2) = a(3n+3) = A208881(n+1). - _R. J. Mathar_, Nov 01 2015
%F A199127 Conjecture: -(458*n-1205) *(n+2) *(n+1)*a(n) +(-208*n^3+2578*n^2-4613*n-2410) *a(n-1) +9*(-339*n-638) *a(n-2) +27*(n-2) *(458*n^2-289*n-1146) *a(n-3) +54*(n-2) *(n-3) *(104*n-1081) *a(n-4)=0. - _R. J. Mathar_, Nov 01 2015
%F A199127 Conjecture: (n+2)*(n+1)*a(n) +(5*n^2-2)*a(n-1) +3*(5*n^2-15*n+3) *a(n-2) +3*(n^2 -60*n +81)*a(n-3) +135*(-n^2+3*n-1)*a(n-4) -405*(n-2)*(n-4) *a(n-5) -810*(n-4) *(n-5) *a(n-6)=0. - _R. J. Mathar_, Nov 01 2015
%F A199127 From _Bob Selcoe_, Sep 26 2021: (Start)
%F A199127 When n == 0 (mod 3), a(n) = n!/(3*(n/3)!^3);
%F A199127 when n == 1 (mod 3), a(n) = n!/(((n+2)/3)!*((n-1)/3)!^2);
%F A199127 when n == 2 (mod 3), a(n) = n!/(((n-2)/3)!*((n+1)/3)!^2).
%F A199127 (End)
%e A199127 Some solutions for n=5:
%e A199127   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1
%e A199127   1 0   1 2   1 2   1 2   1 0   1 2   1 0   1 2   1 2   1 0
%e A199127   0 2   2 0   0 1   2 0   0 2   2 1   0 2   0 1   2 0   2 1
%e A199127   2 1   0 2   2 0   0 1   2 1   1 0   2 0   1 2   0 1   1 2
%e A199127   0 2   2 1   0 2   2 0   1 2   0 2   1 2   2 0   1 2   2 0
%Y A199127 Cf. A286030.
%K A199127 nonn
%O A199127 1,2
%A A199127 _R. H. Hardin_, Nov 03 2011