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%I #8 Dec 08 2018 05:43:56
%S 1,1,1,1,124,3423,33533,158877,490403,1156178,2286874,4013538,6467242,
%T 9779058,14080058,19501314,26173898,34228882,43797338,55010338,
%U 67998954,82894258,99827322,118929218,140331018,164163794,190558618,219646562
%N Number of n X 4 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 R. H. Hardin, <a href="/A253007/b253007.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = (65536/3)*n^3 - 422912*n^2 + (8343128/3)*n - 6208382 for n>9.
%F Conjectures from _Colin Barker_, Dec 08 2018: (Start)
%F G.f.: x*(1 - 3*x + 3*x^2 - x^3 + 123*x^4 + 2930*x^5 + 20582*x^6 + 44788*x^7 + 42525*x^8 + 17119*x^9 + 2605*x^10 + 375*x^11 + 25*x^12) / (1 - x)^4.
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>13.
%F (End)
%e Some solutions for n=6:
%e ..0..0..1..1....0..0..0..0....0..0..0..0....0..1..1..2....0..0..0..1
%e ..0..0..1..1....0..1..1..1....1..1..1..1....0..1..1..2....0..0..0..1
%e ..1..1..1..1....1..1..2..2....1..1..2..2....0..1..1..2....0..0..1..1
%e ..2..2..2..2....1..1..2..2....1..2..2..2....1..1..1..2....0..0..1..2
%e ..2..2..2..2....1..1..2..2....1..2..2..2....1..1..1..2....0..1..1..2
%e ..2..2..2..2....1..1..2..2....1..2..2..2....1..1..1..2....1..1..2..2
%Y Column 4 of A253011.
%K nonn
%O 1,5
%A _R. H. Hardin_, Dec 25 2014