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A212039 Number of lower triangular n X n arrays colored with integers 0 upwards introduced in row major order, with no element equal to any element within a city block distance of two, and containing the value n(n+1)/2-2.

%I A212039 #10 Jul 20 2018 14:47:40
%S A212039 0,1,4,19,58,136,271,484,799,1243,1846,2641,3664,4954,6553,8506,10861,
%T A212039 13669,16984,20863,25366,30556,36499,43264,50923,59551,69226,80029,
%U A212039 92044,105358,120061,136246,154009,173449,194668,217771,242866,270064,299479
%N A212039 Number of lower triangular n X n arrays colored with integers 0 upwards introduced in row major order, with no element equal to any element within a city block distance of two, and containing the value n(n+1)/2-2.
%C A212039 Column 1 of A212044.
%H A212039 R. H. Hardin, <a href="/A212039/b212039.txt">Table of n, a(n) for n = 1..73</a>
%F A212039 Empirical: a(n) = (1/8)*n^4 + (1/4)*n^3 - (25/8)*n^2 + (23/4)*n - 2 for n>1.
%F A212039 Conjectures from _Colin Barker_, Jul 20 2018: (Start)
%F A212039 G.f.: x^2*(1 - x + 9*x^2 - 7*x^3 + x^4) / (1 - x)^5.
%F A212039 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>6.
%F A212039 (End)
%e A212039 Some solutions for n=4:
%e A212039 ..0........0........0........0........0........0........0........0
%e A212039 ..1.2......1.2......1.2......1.2......1.2......1.2......1.2......1.2
%e A212039 ..3.4.5....3.4.5....3.4.5....3.4.5....3.4.1....3.4.5....3.4.5....3.4.5
%e A212039 ..6.7.8.1..6.7.8.0..6.7.3.8..6.7.0.8..5.6.7.8..6.7.8.3..6.7.2.8..6.1.7.8
%Y A212039 Cf. A212044.
%K A212039 nonn
%O A212039 1,3
%A A212039 _R. H. Hardin_, Apr 28 2012