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T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 1 and every value increasing by 0 or 1 with every step right, diagonally se or down
2

%I #4 Dec 24 2014 10:07:11

%S 0,1,1,2,1,2,3,8,8,3,4,16,18,16,4,5,24,94,94,24,5,6,32,190,386,190,32,

%T 6,7,40,290,1729,1729,290,40,7,8,48,390,3577,12200,3577,390,48,8,9,56,

%U 490,5600,51496,51496,5600,490,56,9,10,64,590,7648,109736,608993,109736,7648

%N T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 1 and every value increasing by 0 or 1 with every step right, diagonally se or down

%C Table starts

%C .0..1...2.....3......4........5.........6...........7.............8

%C .1..1...8....16.....24.......32........40..........48............56

%C .2..8..18....94....190......290.......390.........490...........590

%C .3.16..94...386...1729.....3577......5600........7648..........9696

%C .4.24.190..1729..12200....51496....109736......176872........246312

%C .5.32.290..3577..51496...608993...2526480.....5560676.......9241012

%C .6.40.390..5600.109736..2526480..49489706...206094186.....468856938

%C .7.48.490..7648.176872..5560676.206094186..6648891794...28101900201

%C .8.56.590..9696.246312..9241012.468856938.28101900201.1489334202216

%C .9.64.690.11744.316008.13153668.803577258.66058900101.6425402265448

%H R. H. Hardin, <a href="/A252889/b252889.txt">Table of n, a(n) for n = 1..480</a>

%F Empirical for column k:

%F k=1: a(n) = n - 1

%F k=2: a(n) = 8*n - 16 for n>2

%F k=3: a(n) = 100*n - 310 for n>4

%F k=4: a(n) = 2048*n - 8736 for n>6

%F k=5: a(n) = 69696*n - 380952 for n>8

%F k=6: a(n) = 3964928*n - 26499968 for n>10

%F k=7: a(n) = 378224704*n - 2991671254 for n>12

%e Some solutions for n=4 k=4

%e ..0..0..0..1....0..0..0..0....0..0..0..0....0..0..1..1....0..0..1..1

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

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

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

%K nonn,tabl

%O 1,4

%A _R. H. Hardin_, Dec 24 2014