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%I #6 May 12 2023 14:20:24
%S 1,2,2,3,5,4,5,10,13,8,8,18,33,34,16,13,35,70,115,89,32,21,74,154,265,
%T 386,233,64,34,154,433,692,1018,1323,610,128,55,317,1166,2838,3048,
%U 3914,4485,1597,256,89,658,3153,10526,18492,13712,15017,15290,4181,512,144
%N T(n,k) = Number of n X k 0..1 arrays with every element equal to 0, 1, 3 or 4 horizontally or antidiagonally adjacent elements, with upper left element zero.
%C Table starts
%C ...1....2.....3......5.......8.......13........21.........34...........55
%C ...2....5....10.....18......35.......74.......154........317..........658
%C ...4...13....33.....70.....154......433......1166.......3153.........8468
%C ...8...34...115....265.....692.....2838.....10526......36080.......126064
%C ..16...89...386...1018....3048....18492.....94846.....421570......1921284
%C ..32..233..1323...3914...13712...124046....893236....5075048.....30184552
%C ..64..610..4485..15017...61536...829709...8320747...61236964....475553912
%C .128.1597.15290..57850..277248..5574946..78493707..745344973...7586538720
%C .256.4181.51977.222146.1248512.37461858.737846612.9066111545.120905954176
%H R. H. Hardin, <a href="/A302081/b302081.txt">Table of n, a(n) for n = 1..310</a>
%F Empirical for column k:
%F k=1: a(n) = 2*a(n-1).
%F k=2: a(n) = 3*a(n-1) -a(n-2).
%F k=3: a(n) = a(n-1) +7*a(n-2) +4*a(n-3)
%F k=4: a(n) = a(n-1) +14*a(n-2) -a(n-3) -42*a(n-4) -a(n-5) +14*a(n-6) +a(n-7) -a(n-8).
%F k=5: [order 7] for n>9.
%F k=6: [order 32] for n>33.
%F k=7: [order 52] for n>54.
%F Empirical for row n:
%F n=1: a(n) = a(n-1) +a(n-2).
%F n=2: a(n) = 2*a(n-1) +a(n-3) -a(n-4) -2*a(n-6) +a(n-7).
%F n=3: [order 26] for n>27.
%F n=4: [order 92] for n>93.
%e Some solutions for n=5, k=4
%e ..0..1..1..0. .0..1..0..1. .0..1..1..1. .0..1..0..1. .0..0..1..0
%e ..0..0..1..0. .0..0..1..1. .0..1..0..1. .0..1..1..1. .1..0..0..1
%e ..1..0..0..1. .0..1..0..0. .0..0..0..1. .0..1..0..1. .1..0..1..0
%e ..1..0..1..0. .0..1..1..1. .1..1..0..1. .0..1..0..1. .1..0..1..0
%e ..1..0..1..1. .0..1..0..0. .0..0..0..1. .0..1..0..0. .0..1..0..1
%Y Column 1 is A000079(n-1).
%Y Column 2 is A001519(n+1).
%Y Row 1 is A000045(n+1).
%Y Row 2 is A301885.
%K nonn,tabl
%O 1,2
%A _R. H. Hardin_, Mar 31 2018
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