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T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 0, 2 or 4 king-move neighboring 1s.
8

%I #4 Dec 19 2017 13:18:08

%S 2,3,3,5,9,5,8,21,21,8,13,57,69,57,13,21,153,258,258,153,21,34,393,

%T 963,1463,963,393,34,55,1041,3493,8315,8315,3493,1041,55,89,2745,

%U 12860,44668,72533,44668,12860,2745,89,144,7185,47305,246923,579857,579857,246923

%N T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 0, 2 or 4 king-move neighboring 1s.

%C Table starts

%C ..2....3......5.......8........13..........21...........34.............55

%C ..3....9.....21......57.......153.........393.........1041...........2745

%C ..5...21.....69.....258.......963........3493........12860..........47305

%C ..8...57....258....1463......8315.......44668.......246923........1364492

%C .13..153....963....8315.....72533......579857......4846823.......40521141

%C .21..393...3493...44668....579857.....6829945.....84312938.....1042553987

%C .34.1041..12860..246923...4846823....84312938...1554755519....28740762011

%C .55.2745..47305.1364492..40521141..1042553987..28740762011...795154816254

%C .89.7185.173498.7488185.334815607.12735280455.523224064059.21598350633016

%H R. H. Hardin, <a href="/A296725/b296725.txt">Table of n, a(n) for n = 1..238</a>

%F Empirical for column k:

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

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

%F k=3: [order 10]

%F k=4: [order 18]

%F k=5: [order 45]

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A000045(n+2).

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Dec 19 2017