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T(n,k)=Number of length n+2 0..k arrays with no three unequal elements in a row and no three equal elements in a row and new values 0..k introduced in 0..k order
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%I #4 Jun 09 2014 11:07:43

%S 3,3,5,3,6,8,3,6,12,13,3,6,12,24,21,3,6,12,25,48,34,3,6,12,25,53,96,

%T 55,3,6,12,25,53,115,192,89,3,6,12,25,53,116,253,384,144,3,6,12,25,53,

%U 116,259,564,768,233,3,6,12,25,53,116,259,593,1268,1536,377,3,6,12,25,53,116,259

%N T(n,k)=Number of length n+2 0..k arrays with no three unequal elements in a row and no three equal elements in a row and new values 0..k introduced in 0..k order

%C Table starts

%C ...3....3....3....3....3....3....3....3....3....3....3....3....3....3....3....3

%C ...5....6....6....6....6....6....6....6....6....6....6....6....6....6....6....6

%C ...8...12...12...12...12...12...12...12...12...12...12...12...12...12...12...12

%C ..13...24...25...25...25...25...25...25...25...25...25...25...25...25...25...25

%C ..21...48...53...53...53...53...53...53...53...53...53...53...53...53...53...53

%C ..34...96..115..116..116..116..116..116..116..116..116..116..116..116..116..116

%C ..55..192..253..259..259..259..259..259..259..259..259..259..259..259..259..259

%C ..89..384..564..593..594..594..594..594..594..594..594..594..594..594..594..594

%C .144..768.1268.1382.1389.1389.1389.1389.1389.1389.1389.1389.1389.1389.1389.1389

%C .233.1536.2871.3281.3322.3323.3323.3323.3323.3323.3323.3323.3323.3323.3323.3323

%H R. H. Hardin, <a href="/A243729/b243729.txt">Table of n, a(n) for n = 1..9999</a>

%F Empirical for column k:

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

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

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

%F k=4: a(n) = 4*a(n-1) -13*a(n-3) +6*a(n-4) +8*a(n-5)

%F k=5: a(n) = 5*a(n-1) -29*a(n-3) +20*a(n-4) +51*a(n-5) -31*a(n-6) -30*a(n-7)

%F k=6: [order 9]

%F k=7: [order 11]

%F k=8: [order 13]

%F k=9: [order 15]

%e Some solutions for n=6 k=4

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

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

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

%e ..1....1....0....0....0....2....1....1....2....2....2....2....0....0....1....1

%e ..0....2....1....1....0....2....0....1....2....2....2....1....2....1....0....0

%e ..0....2....1....0....2....3....1....2....0....3....1....2....2....0....0....1

%e ..1....3....2....1....2....3....1....1....0....2....1....2....3....0....2....1

%e ..0....2....1....1....0....0....0....2....1....3....2....0....3....2....0....2

%Y Column 1 is A000045(n+3)

%Y Column 2 is A003945

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Jun 09 2014