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T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 4, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
12

%I #4 Apr 21 2018 12:48:28

%S 1,2,2,3,3,4,5,3,4,8,8,5,12,6,16,13,7,17,11,9,32,21,13,24,36,19,14,64,

%T 34,23,67,50,74,34,22,128,55,37,158,128,139,165,53,35,256,89,63,298,

%U 439,410,349,361,83,56,512,144,109,595,1085,1799,1221,853,783,136,90,1024,233

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 4, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

%C Table starts

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

%C ...2..3...3....5....7....13.....23......37.......63.......109........183

%C ...4..4..12...17...24....67....158.....298......595......1337.......2863

%C ...8..6..11...36...50...128....439....1085.....2431......6452......17455

%C ..16..9..19...74..139...410...1799....5907....16494.....53290.....184915

%C ..32.14..34..165..349..1221...7096...30280...102683....403872....1783894

%C ..64.22..53..361..853..3453..26184..148313...618149...2955145...16591424

%C .128.35..83..783.2180.10223.100128..746323..3851318..22515378..159560449

%C .256.56.136.1710.5525.30247.387892.3784002.23967605.171306353.1539204838

%H R. H. Hardin, <a href="/A303314/b303314.txt">Table of n, a(n) for n = 1..511</a>

%F Empirical for column k:

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

%F k=2: a(n) = 2*a(n-1) -a(n-3)

%F k=3: a(n) = a(n-1) +a(n-3) +a(n-4) for n>7

%F k=4: a(n) = a(n-1) +a(n-2) +3*a(n-3) +2*a(n-4) -a(n-5) -2*a(n-6) -a(n-7) for n>10

%F k=5: a(n) = a(n-1) +9*a(n-3) +2*a(n-4) +4*a(n-5) -10*a(n-6) -6*a(n-7) +4*a(n-9) for n>12

%F k=6: [order 8] for n>11

%F k=7: [order 20] for n>23

%F Empirical for row n:

%F n=1: a(n) = a(n-1) +a(n-2)

%F n=2: a(n) = a(n-1) +2*a(n-3) for n>5

%F n=3: a(n) = a(n-1) +3*a(n-3) +4*a(n-4) for n>7

%F n=4: a(n) = a(n-1) +a(n-2) +5*a(n-3) +9*a(n-4) -3*a(n-5) -7*a(n-6) -2*a(n-7) for n>11

%F n=5: [order 12] for n>16

%F n=6: [order 23] for n>28

%F n=7: [order 46] for n>51

%e Some solutions for n=5 k=4

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

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

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

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

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

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

%Y Column 2 is A001611(n+1).

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

%Y Row 2 is A003229(n-1) for n>2.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Apr 21 2018