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Number of lower triangular n X n arrays colored with integers 0 upwards introduced in row major order, with no element equal to any horizontal, vertical or antidiagonal neighbor, and containing the value n(n+1)/2-2
2

%I #9 Mar 09 2018 08:32:35

%S 0,2,9,31,80,171,322,554,891,1360,1991,2817,3874,5201,6840,8836,11237,

%T 14094,17461,21395,25956,31207,37214,44046,51775,60476,70227,81109,

%U 93206,106605,121396,137672,155529,175066,196385,219591,244792,272099,301626

%N Number of lower triangular n X n arrays colored with integers 0 upwards introduced in row major order, with no element equal to any horizontal, vertical or antidiagonal neighbor, and containing the value n(n+1)/2-2

%C Column 1 of A211930.

%H R. H. Hardin, <a href="/A211924/b211924.txt">Table of n, a(n) for n = 1..74</a>

%F Empirical: a(n) = (1/8)*n^4 + (1/4)*n^3 - (13/8)*n^2 + (9/4)*n for n>1.

%F Conjectures from _Colin Barker_, Mar 09 2018: (Start)

%F G.f.: x^2*(2 - x)*(1 + 3*x^2 - x^3) / (1 - x)^5.

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.

%F (End)

%e Some solutions for n=4:

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

%e ..1.2......1.2......1.0......1.2......1.2......1.2......1.2......1.2

%e ..3.4.5....3.4.5....2.3.4....3.0.4....3.4.5....0.3.4....3.4.5....3.4.3

%e ..6.0.7.8..6.2.7.8..5.6.7.8..5.6.7.8..6.7.8.4..5.6.7.8..6.7.8.7..5.6.7.8

%Y Cf. A211930.

%K nonn

%O 1,2

%A _R. H. Hardin_, Apr 25 2012