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T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1 or 3 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.
8

%I #4 Aug 06 2018 12:14:20

%S 1,2,2,3,3,3,5,4,4,5,8,7,5,7,8,13,11,11,11,11,13,21,18,16,10,16,18,21,

%T 34,30,28,14,14,28,30,34,55,49,44,23,19,23,44,49,55,89,81,74,32,30,30,

%U 32,74,81,89,144,134,122,55,42,40,42,55,122,134,144,233,221,209,86,67,54,54,67

%N T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1 or 3 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

%C Table starts

%C ..1..2...3..5...8..13..21..34..55..89.144.233..377..610..987.1597.2584..4181

%C ..2..3...4..7..11..18..30..49..81.134.221.365..603..996.1645.2717.4488..7413

%C ..3..4...5.11..16..28..44..74.122.209.344.584..981.1670.2806.4758.8047.13665

%C ..5..7..11.10..14..23..32..55..86.143.226.367..599..967.1581.2574.4220..6868

%C ..8.11..16.14..19..30..42..67.100.160.248.401..630.1025.1615.2605.4136..6726

%C .13.18..28.23..30..40..54..82.115.185.266.425..649.1028.1588.2487.3908..6096

%C .21.30..44.32..42..54..75.107.149.221.317.491..739.1157.1774.2819.4306..6792

%C .34.49..74.55..67..82.107.159.209.296.408.604..862.1336.1951.3069.4613..7156

%C .55.81.122.86.100.115.149.209.310.405.554.784.1110.1634.2337.3542.5297..8100

%H R. H. Hardin, <a href="/A317773/b317773.txt">Table of n, a(n) for n = 1..1098</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-3) -a(n-6)

%F k=3: [order 17] for n>23

%F k=4: [order 14] for n>23

%F k=5: [order 33] for n>41

%F k=6: [order 65] for n>72

%e Some solutions for n=5 k=4

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

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

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

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

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

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

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Aug 06 2018