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T(n,k)=Number of nXk 0..1 arrays with every repeated value in every row greater than or equal to, and in every column greater than, the previous repeated value.
12

%I #4 Jan 20 2016 12:31:18

%S 2,4,4,8,16,6,15,64,36,9,28,225,216,81,12,51,784,1056,729,144,16,92,

%T 2601,5004,5081,1728,256,20,164,8464,22110,34173,14956,4096,400,25,

%U 290,26896,94554,211555,122770,44742,8000,625,30,509,84100,391314,1262760,912667

%N T(n,k)=Number of nXk 0..1 arrays with every repeated value in every row greater than or equal to, and in every column greater than, the previous repeated value.

%C Table starts

%C ..2....4.....8.....15.......28........51.........92.........164..........290

%C ..4...16....64....225......784......2601.......8464.......26896........84100

%C ..6...36...216...1056.....5004.....22110......94554......391314......1582824

%C ..9...81...729...5081....34173....211555....1262760.....7263481.....40755550

%C .12..144..1728..14956...122770....912667....6484282....44116906....291598056

%C .16..256..4096..44742...460598...4245574...37282358...312449872...2540944329

%C .20..400..8000.102954..1234716..13126812..132388406..1271314080..11831791048

%C .25..625.15625.238813..3380133..42012357..494152778..5520112546..59723941668

%C .30..900.27000.472174..7591852.106570618.1413416776.17806098826.217360101006

%C .36.1296.46656.935890.17155354.272337497.4075463059.57801662876.793861159136

%H R. H. Hardin, <a href="/A267788/b267788.txt">Table of n, a(n) for n = 1..200</a>

%F Empirical for column k:

%F k=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4)

%F k=2: a(n) = 2*a(n-1) +2*a(n-2) -6*a(n-3) +6*a(n-5) -2*a(n-6) -2*a(n-7) +a(n-8)

%F k=3: [order 12]

%F k=4: [order 16] for n>18

%F k=5: [order 20] for n>22

%F k=6: [order 24] for n>27

%F Empirical for row n:

%F n=1: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) -a(n-4)

%F n=2: [order 9]

%F n=3: [order 12]

%F n=4: [order 93]

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 1 is A002620(n+2).

%Y Column 2 is A030179(n+2).

%Y Row 1 is A029907(n+1).

%Y Row 2 is A267729.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Jan 20 2016