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T(n,k) = Number of n X k 0..4 arrays with no element equal to another at a city block distance of exactly two, and new values 0..4 introduced in row major order.
6

%I #6 Apr 26 2021 21:02:16

%S 1,2,2,3,7,3,7,33,33,7,20,273,420,273,20,66,2433,5880,5880,2433,66,

%T 238,21873,82320,130680,82320,21873,238,902,196833,1152480,2883000,

%U 2883000,1152480,196833,902,3510,1771473,16134720,63772920,100692120,63772920

%N T(n,k) = Number of n X k 0..4 arrays with no element equal to another at a city block distance of exactly two, and new values 0..4 introduced in row major order.

%C Table starts

%C ......1..........2...........3............7............20.............66

%C ......2..........7..........33..........273..........2433..........21873

%C ......3.........33.........420.........5880.........82320........1152480

%C ......7........273........5880.......130680.......2883000.......63772920

%C .....20.......2433.......82320......2883000.....100692120.....3510873720

%C .....66......21873.....1152480.....63772920....3510873720...194281940280

%C ....238.....196833....16134720...1409319480..122511965400.10720371319680

%C ....902....1771473...225886080..31155452280.4273149243000

%C ...3510...15943233..3162405120.688658403000

%C ..13846..143489073.44273671680

%C ..54998.1291401633

%C .219222

%H R. H. Hardin, <a href="/A222944/b222944.txt">Table of n, a(n) for n = 1..84</a>

%F Empirical for column k:

%F k=1: a(n) = 7*a(n-1) -14*a(n-2) +8*a(n-3) for n > 5;

%F k=2: a(n) = 10*a(n-1) -9*a(n-2) for n > 4;

%F k=3: a(n) = 14*a(n-1) for n > 3;

%F k=4: a(n) = 17*a(n-1) +136*a(n-2) -512*a(n-3) for n > 5;

%F k=5: [order 16 for n > 18].

%e Some solutions for n=3, k=4

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

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

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

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_ Mar 09 2013