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T(n,k) = Number of n X k 0..1 arrays with every element equal to 0, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero.
7

%I #6 Apr 17 2022 22:09:44

%S 1,1,1,1,5,1,1,12,12,1,1,37,22,37,1,1,104,81,81,104,1,1,301,307,427,

%T 307,301,1,1,864,1201,2338,2338,1201,864,1,1,2485,5066,13458,21730,

%U 13458,5066,2485,1,1,7144,21292,84948,202841,202841,84948,21292,7144,1,1,20541

%N T(n,k) = Number of n X k 0..1 arrays with every element equal to 0, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero.

%C Table starts

%C .1....1.....1.......1.........1...........1............1..............1

%C .1....5....12......37.......104.........301..........864...........2485

%C .1...12....22......81.......307........1201.........5066..........21292

%C .1...37....81.....427......2338.......13458........84948.........543741

%C .1..104...307....2338.....21730......202841......1992466.......19685956

%C .1..301..1201...13458....202841.....3096833.....48911434......775504649

%C .1..864..5066...84948...1992466....48911434...1226106440....30729398000

%C .1.2485.21292..543741..19685956...775504649..30729398000..1213390065190

%C .1.7144.90443.3534493.195564094.12397088059.778116037031.48505143319432

%H R. H. Hardin, <a href="/A299221/b299221.txt">Table of n, a(n) for n = 1..180</a>

%F Empirical for column k:

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

%F k=2: a(n) = 5*a(n-2) +8*a(n-3) +4*a(n-4),

%F k=3: [order 19] for n>20,

%F k=4: [order 66] for n>68.

%e Some solutions for n=5, k=4

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

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

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

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

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

%Y Column 2 is A297909.

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Feb 05 2018