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%I #4 Mar 29 2018 12:46:02
%S 0,1,0,1,2,0,2,5,5,0,3,16,20,13,0,5,52,123,83,34,0,8,169,680,947,342,
%T 89,0,13,549,4070,9084,7326,1411,233,0,21,1784,23565,98839,120815,
%U 56710,5820,610,0,34,5797,138014,1029960,2406169,1608681,439078,24007,1597,0
%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2 or 3 horizontally or antidiagonally adjacent elements, with upper left element zero.
%C Table starts
%C .0....1.....1........2..........3............5...............8
%C .0....2.....5.......16.........52..........169.............549
%C .0....5....20......123........680.........4070...........23565
%C .0...13....83......947.......9084........98839.........1029960
%C .0...34...342.....7326.....120815......2406169........45013365
%C .0...89..1411....56710....1608681.....58609226......1969215107
%C .0..233..5820...439078...21418808...1427656268.....86143630040
%C .0..610.24007..3399722..285190208..34776685046...3768464135104
%C .0.1597.99026.26323903.3797277789.847137052736.164856325277648
%H R. H. Hardin, <a href="/A301951/b301951.txt">Table of n, a(n) for n = 1..364</a>
%F Empirical for column k:
%F k=1: a(n) = a(n-1)
%F k=2: a(n) = 3*a(n-1) -a(n-2)
%F k=3: a(n) = 4*a(n-1) +a(n-2) -2*a(n-3)
%F k=4: a(n) = 9*a(n-1) -8*a(n-2) -14*a(n-3) +4*a(n-4) +4*a(n-5) -a(n-6)
%F k=5: [order 13] for n>15
%F k=6: [order 26] for n>28
%F k=7: [order 43] for n>47
%F Empirical for row n:
%F n=1: a(n) = a(n-1) +a(n-2)
%F n=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4) for n>6
%F n=3: [order 10] for n>12
%F n=4: [order 36] for n>40
%e Some solutions for n=5 k=4
%e ..0..0..0..0. .0..0..0..0. .0..0..1..1. .0..0..0..1. .0..0..1..1
%e ..0..1..1..0. .0..1..1..0. .0..1..1..1. .0..0..1..1. .0..1..0..1
%e ..0..0..0..0. .1..1..0..0. .0..0..1..1. .0..0..1..0. .0..0..1..0
%e ..1..1..1..1. .1..0..0..1. .1..1..1..1. .0..0..0..0. .1..1..0..0
%e ..0..0..0..0. .1..1..1..1. .0..0..0..0. .1..1..1..1. .1..1..1..1
%Y Column 2 is A001519.
%Y Row 1 is A000045(n-1).
%Y Row 2 is A232317(n-1).
%K nonn,tabl
%O 1,5
%A _R. H. Hardin_, Mar 29 2018
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