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Number of n X 5 nonnegative integer arrays with upper left 0 and every value within 3 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.
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%I #8 Dec 07 2018 11:51:50

%S 15,83,494,3067,17962,86488,320270,917811,2127013,4211511,7437417,

%T 12070971,18378413,26625983,37079921,50006467,65671861,84342343,

%U 106284153,131763531,161046717,194399951,232089473,274381523,321542341

%N Number of n X 5 nonnegative integer arrays with upper left 0 and every value within 3 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

%H R. H. Hardin, <a href="/A252935/b252935.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (133120/3)*n^3 - 760496*n^2 + (13526246/3)*n - 9199709 for n>8.

%F Conjectures from _Colin Barker_, Dec 07 2018: (Start)

%F G.f.: x*(15 + 23*x + 252*x^2 + 1529*x^3 + 8341*x^4 + 31149*x^5 + 70316*x^6 + 86878*x^7 + 49399*x^8 + 15733*x^9 + 2477*x^10 + 128*x^11) / (1 - x)^4.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>12.

%F (End)

%e Some solutions for n=4:

%e ..0..1..2..2..3....0..0..1..1..1....0..1..1..2..2....0..1..1..1..1

%e ..1..1..2..2..3....0..1..1..1..1....0..1..2..2..3....0..1..1..1..2

%e ..1..1..2..3..3....0..1..2..2..2....1..1..2..3..3....0..1..1..1..2

%e ..1..1..2..3..4....1..1..2..3..3....1..2..2..3..4....0..1..2..2..2

%Y Column 5 of A252938.

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 24 2014