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T(n,k) = Number of n X k 0..2 arrays with no element unequal to more than four of its king-move neighbors and with new values introduced in order 0 sequentially upwards.
6

%I #6 May 03 2021 12:59:42

%S 1,2,2,5,14,5,14,91,91,14,41,600,388,600,41,122,4039,2754,2754,4039,

%T 122,365,27212,15543,31752,15543,27212,365,1094,183195,95045,237031,

%U 237031,95045,183195,1094,3281,1233336,565516,2109626,2091092,2109626,565516

%N T(n,k) = Number of n X k 0..2 arrays with no element unequal to more than four of its king-move neighbors and with new values introduced in order 0 sequentially upwards.

%C Table starts

%C ....1........2.........5..........14...........41..........122...........365

%C ....2.......14........91.........600.........4039........27212........183195

%C ....5.......91.......388........2754........15543........95045........565516

%C ...14......600......2754.......31752.......237031......2109626......17495647

%C ...41.....4039.....15543......237031......2091092.....22779237.....235194056

%C ..122....27212.....95045.....2109626.....22779237....299676213....3900134833

%C ..365...183195....565516....17495647....235194056...3900134833...64606412468

%C .1094..1233336...3403942...148941378...2461439740..50732580694.1061812396215

%C .3281..8303647..20376721..1259597639..25939234980.672441825967

%C .9842.55905604.122289647.10659570010.271794661137

%H R. H. Hardin, <a href="/A282056/b282056.txt">Table of n, a(n) for n = 1..112</a>

%F Empirical for column k:

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

%F k=2: a(n) = 8*a(n-1) -11*a(n-2) +20*a(n-3) -24*a(n-4) +8*a(n-5).

%F k=3: [order 16] for n > 18.

%F k=4: [order 41] for n > 45.

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

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

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

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

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

%Y Column 1 is A007051(n-1).

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Feb 05 2017