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A252836 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 2 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.

%I A252836 #6 Jul 23 2025 13:41:13
%S A252836 1,2,2,4,5,4,7,13,13,7,11,29,44,29,11,16,53,127,127,53,16,22,85,288,
%T A252836 493,288,85,22,29,125,529,1474,1474,529,125,29,37,173,850,3365,6068,
%U A252836 3365,850,173,37,46,229,1251,6211,18528,18528,6211,1251,229,46,56,293,1732,10017
%N A252836 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 2 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.
%C A252836 Table starts
%C A252836 ..1...2....4.....7.....11......16.......22........29........37.........46
%C A252836 ..2...5...13....29.....53......85......125.......173.......229........293
%C A252836 ..4..13...44...127....288.....529......850......1251......1732.......2293
%C A252836 ..7..29..127...493...1474....3365.....6211.....10017.....14783......20509
%C A252836 .11..53..288..1474...6068...18528....42738.....79563....129173.....191583
%C A252836 .16..85..529..3365..18528...78674...244131....569420...1070036....1750150
%C A252836 .22.125..850..6211..42738..244131..1056756...3320837...7822881...14827629
%C A252836 .29.173.1251.10017..79563..569420..3320837..14564701..46222275..109796227
%C A252836 .37.229.1732.14783.129173.1070036..7822881..46222275.204666202..654583926
%C A252836 .46.293.2293.20509.191583.1750150.14827629.109796227.654583926.2919462498
%H A252836 R. H. Hardin, <a href="/A252836/b252836.txt">Table of n, a(n) for n = 1..1135</a>
%F A252836 Empirical for column k:
%F A252836 k=1: a(n) = (1/2)*n^2 - (1/2)*n + 1
%F A252836 k=2: a(n) = 4*n^2 - 12*n + 13 for n>1
%F A252836 k=3: a(n) = 40*n^2 - 199*n + 283 for n>3
%F A252836 k=4: a(n) = 480*n^2 - 3394*n + 6449 for n>5
%F A252836 k=5: a(n) = 6400*n^2 - 59190*n + 143483 for n>7
%F A252836 k=6: a(n) = 90112*n^2 - 1032064*n + 3059590 for n>9
%F A252836 k=7: a(n) = 1306624*n^2 - 17846996*n + 62638467 for n>11
%e A252836 Some solutions for n=4 k=4
%e A252836 ..0..1..2..2....0..1..1..2....0..0..1..1....0..0..1..1....0..0..1..1
%e A252836 ..1..1..2..2....0..1..1..2....1..1..1..1....0..1..1..1....1..1..1..1
%e A252836 ..1..1..2..3....1..1..2..2....1..1..2..2....0..1..1..1....1..1..1..1
%e A252836 ..1..2..2..3....1..1..2..2....1..2..2..3....1..1..2..2....1..1..2..2
%Y A252836 Column 1 is A000124(n-1)
%Y A252836 Column 2 is A078370(n-2)
%K A252836 nonn,tabl
%O A252836 1,2
%A A252836 _R. H. Hardin_, Dec 22 2014