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%I #9 Dec 06 2018 19:24:52
%S 4,18,81,340,1238,3891,10761,26764,60988,129236,257653,487744,883142,
%T 1538541,2591269,4236040,6743492,10483190,15951849,23807612,34911302,
%U 50375655,71623633,100457012,139136540,190475064,257945133,345802696
%N Number of n X 3 nonnegative integer arrays with upper left 0 and every value within 3 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.
%H R. H. Hardin, <a href="/A252823/b252823.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = (1/90720)*n^9 + (1/2520)*n^8 + (19/3780)*n^7 + (1/45)*n^6 + (157/4320)*n^5 + (31/180)*n^4 + (28507/45360)*n^3 + (769/2520)*n^2 + (1783/630)*n.
%F Conjectures from _Colin Barker_, Dec 06 2018: (Start)
%F G.f.: x*(4 - 22*x + 81*x^2 - 140*x^3 + 163*x^4 - 137*x^5 + 75*x^6 - 23*x^7 + 3*x^8) / (1 - x)^10.
%F a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>10.
%F (End)
%e Some solutions for n=4:
%e ..0..0..0....0..0..1....0..0..1....0..0..1....0..0..0....0..0..0....0..0..1
%e ..0..1..1....1..1..2....0..1..2....1..1..2....0..1..1....0..0..0....0..0..1
%e ..1..2..2....2..2..3....1..1..2....2..2..3....0..1..2....0..0..1....1..1..2
%e ..2..2..3....3..3..4....1..1..2....2..3..3....1..2..3....0..1..2....1..2..3
%Y Column 3 of A252828.
%K nonn
%O 1,1
%A _R. H. Hardin_, Dec 22 2014