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A253006 Number of n X 3 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 3 and every value increasing by 0 or 1 with every step right, diagonally se or down.

%I A253006 #8 Dec 08 2018 05:28:47
%S A253006 0,0,0,1,54,632,2902,8416,18770,35564,60398,94872,140586,199140,
%T A253006 272134,361168,467842,593756,740510,909704,1102938,1321812,1567926,
%U A253006 1842880,2148274,2485708,2856782,3263096,3706250,4187844,4709478,5272752,5879266
%N A253006 Number of n X 3 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 3 and every value increasing by 0 or 1 with every step right, diagonally se or down.
%H A253006 R. H. Hardin, <a href="/A253006/b253006.txt">Table of n, a(n) for n = 1..210</a>
%F A253006 Empirical: a(n) = (800/3)*n^3 - 3980*n^2 + (60442/3)*n - 34576 for n>6.
%F A253006 Conjectures from _Colin Barker_, Dec 08 2018: (Start)
%F A253006 G.f.: x^4*(1 + 50*x + 422*x^2 + 694*x^3 + 385*x^4 + 44*x^5 + 4*x^6) / (1 - x)^4.
%F A253006 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>10.
%F A253006 (End)
%e A253006 Some solutions for n=6:
%e A253006 ..0..0..0....0..0..0....0..1..2....0..1..2....0..0..1....0..0..0....0..0..1
%e A253006 ..0..0..0....1..1..1....1..1..2....1..1..2....0..1..1....0..0..1....0..0..1
%e A253006 ..0..0..1....1..1..1....1..2..2....1..2..2....1..1..2....1..1..1....0..1..1
%e A253006 ..0..1..1....1..1..1....1..2..2....1..2..2....2..2..2....1..1..2....1..1..2
%e A253006 ..0..1..1....1..2..2....1..2..2....2..2..2....2..2..2....1..2..2....1..1..2
%e A253006 ..1..1..2....1..2..2....1..2..2....2..2..2....2..2..2....2..2..2....1..1..2
%Y A253006 Column 3 of A253011.
%K A253006 nonn
%O A253006 1,5
%A A253006 _R. H. Hardin_, Dec 25 2014