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T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero.
6

%I #4 Jun 23 2018 12:53:19

%S 1,2,2,3,5,3,5,9,9,5,8,21,13,21,8,13,53,30,30,53,13,21,105,66,93,66,

%T 105,21,34,237,123,249,249,123,237,34,55,577,252,544,832,544,252,577,

%U 55,89,1205,535,1372,1956,1956,1372,535,1205,89,144,2681,1074,3411,5421,5277

%N T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero.

%C Table starts

%C ..1....2....3....5.....8.....13.....21......34.......55.......89.......144

%C ..2....5....9...21....53....105....237.....577.....1205.....2681......6349

%C ..3....9...13...30....66....123....252.....535.....1074.....2194......4530

%C ..5...21...30...93...249....544...1372....3411.....8269....20472.....50397

%C ..8...53...66..249...832...1956...5421...15993....42682...118773....341634

%C .13..105..123..544..1956...5277..16195...52752...158458...503426...1626380

%C .21..237..252.1372..5421..16195..53834..185090...619073..2213741...7903854

%C .34..577..535.3411.15993..52752.185090..722779..2706166.10841923..44886700

%C .55.1205.1074.8269.42682.158458.619073.2706166.11592019.54162509.259464240

%H R. H. Hardin, <a href="/A306172/b306172.txt">Table of n, a(n) for n = 1..241</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) +8*a(n-3) -4*a(n-4)

%F k=3: a(n) = a(n-1) +4*a(n-3) +2*a(n-5) -a(n-7) -2*a(n-9) -2*a(n-11) -a(n-12)

%F k=4: [order 40] for n>41

%e Some solutions for n=5 k=4

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

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

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

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

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

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

%Y Column 2 is A303963.

%Y Column 3 is A304664.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Jun 23 2018