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T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 5, 6 or 8 king-move adjacent elements, with upper left element zero.
7

%I #4 Feb 18 2018 10:33:17

%S 1,2,2,4,8,4,8,26,26,8,16,88,95,88,16,32,298,375,375,298,32,64,1012,

%T 1524,1998,1524,1012,64,128,3440,6170,11733,11733,6170,3440,128,256,

%U 11700,24838,64703,115279,64703,24838,11700,256,512,39804,100272,358219,915181

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 5, 6 or 8 king-move adjacent elements, with upper left element zero.

%C Table starts

%C ...1.....2......4........8........16..........32...........64............128

%C ...2.....8.....26.......88.......298........1012.........3440..........11700

%C ...4....26.....95......375......1524........6170........24838.........100272

%C ...8....88....375.....1998.....11733.......64703.......358219........2010841

%C ..16...298...1524....11733....115279......915181......7443583.......65049552

%C ..32..1012...6170....64703....915181.....9822603....109073249.....1317338135

%C ..64..3440..24838...358219...7443583...109073249...1686794332....28696396155

%C .128.11700.100272..2010841..65049552..1317338135..28696396155...723176599097

%C .256.39804.405068.11248655.548247606.15245980679.463029637815.16724897634952

%H R. H. Hardin, <a href="/A299753/b299753.txt">Table of n, a(n) for n = 1..180</a>

%F Empirical for column k:

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

%F k=2: a(n) = 4*a(n-1) -2*a(n-2) +2*a(n-3) -6*a(n-4) -4*a(n-5)

%F k=3: [order 18] for n>19

%F k=4: [order 65] for n>66

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A000079(n-1).

%Y Column 2 is A298189.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Feb 18 2018