%I #13 May 23 2018 20:03:57

%S 0,4,4,10,16,26,40,59,84,116,156,205,264,334,416,511,620,744,884,1041,

%T 1216,1410,1624,1859,2116,2396,2700,3029,3384,3766,4176,4615,5084,

%U 5584,6116,6681,7280,7914,8584,9291,10036,10820,11644,12509,13416,14366,15360

%N Number of n X 1 0..3 arrays with rows and columns lexicographically nondecreasing and every element equal to at least one horizontal or vertical neighbor.

%C Column 1 of A201625.

%H R. H. Hardin, <a href="/A201618/b201618.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (1/6)*n^3 - n^2 + (35/6)*n - 9 for n>4.

%F For n > 4 the above empirical a(n) is equal to C(n-1,3) + 4C(n-2,1) that is the n-th coefficient in Taylor series of ((1-x+x^2)/(1-x))^4 at x=0. - _Nikita Gogin_, Jul 24 2013

%F Conjectures from _Colin Barker_, May 23 2018: (Start)

%F G.f.: x^2*(2 - 2*x + x^2)*(2 - 4*x + 4*x^2 - 2*x^3 + x^4) / (1 - x)^4.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>8.

%F (End)

%e Some solutions for n=10:

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

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

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

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

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

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

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

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

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

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

%K nonn

%O 1,2

%A _R. H. Hardin_, Dec 03 2011