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T(n,k) = Number of length n+2 0..k arrays with no three elements in a row with pattern aba (with a!=b) and new values 0..k introduced in 0..k order.
9

%I #6 Apr 17 2022 22:10:45

%S 3,4,5,4,9,8,4,10,21,13,4,10,28,50,21,4,10,29,85,120,34,4,10,29,96,

%T 269,289,55,4,10,29,97,349,870,697,89,4,10,29,97,365,1350,2844,1682,

%U 144,4,10,29,97,366,1511,5425,9346,4060,233,4,10,29,97,366,1533,6726,22297,30792

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

%C Table starts

%C ...3....4......4......4......4......4.......4.......4.......4.......4.......4

%C ...5....9.....10.....10.....10.....10......10......10......10......10......10

%C ...8...21.....28.....29.....29.....29......29......29......29......29......29

%C ..13...50.....85.....96.....97.....97......97......97......97......97......97

%C ..21..120....269....349....365....366.....366.....366.....366.....366.....366

%C ..34..289....870...1350...1511...1533....1534....1534....1534....1534....1534

%C ..55..697...2844...5425...6726...7020....7049....7050....7050....7050....7050

%C ..89.1682...9346..22297..31544..34631...35129...35166...35167...35167...35167

%C .144.4060..30792..92841.153328.181391..187993..188788..188834..188835..188835

%C .233.9801.101577.389456.763187.995139.1069914.1082913.1084123.1084179.1084180

%H R. H. Hardin, <a href="/A243607/b243607.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) = 3*a(n-1) -a(n-2) -a(n-3),

%F k=3: a(n) = 5*a(n-1) -5*a(n-2) -3*a(n-3) +3*a(n-4) +a(n-5),

%F k=4: a(n) = 8*a(n-1) -18*a(n-2) +5*a(n-3) +17*a(n-4) -4*a(n-5) -6*a(n-6) -a(n-7),

%F k=5: [order 9],

%F k=6: [order 11],

%F k=7: [order 13],

%F k=8: [order 15],

%F k=9: [order 17].

%e Some solutions for n=5, k=4

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

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

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

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

%e ..2....2....1....3....3....1....0....1....2....0....3....4....1....1....2....1

%e ..3....3....0....1....3....1....4....1....2....0....3....0....3....3....2....1

%e ..4....1....2....1....0....0....3....0....0....2....3....1....0....3....1....0

%Y Column 1 is A000045(n+3).

%Y Column 2 is A024537(n+1).

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Jun 07 2014