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Number of 0..4 arrays of length n with no adjacent pair equal to its immediately preceding adjacent pair, and new values introduced in 0..4 order.
1

%I #8 Jul 21 2018 10:06:34

%S 1,2,5,13,44,167,695,3070,14074,65958,313098,1497216,7189646,34606966,

%T 166803484,804596882,3882748894,18741593296,90476092366,436812623774,

%U 2108996095916,10182801139146,49166003981046,237391983175872

%N Number of 0..4 arrays of length n with no adjacent pair equal to its immediately preceding adjacent pair, and new values introduced in 0..4 order.

%C Column 4 of A212829.

%H R. H. Hardin, <a href="/A212825/b212825.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 7*a(n-1) - 7*a(n-2) - 20*a(n-3) + 10*a(n-4) + 24*a(n-5) + 8*a(n-6) for n>9.

%F Empirical g.f.: x*(1 + x + x^2)*(1 - 6*x + 3*x^2 + 15*x^3 - 9*x^5 - 3*x^6) / ((1 - x - x^2)*(1 - 2*x - 2*x^2)*(1 - 4*x - 4*x^2)). - _Colin Barker_, Jul 21 2018

%e Some solutions for n=8:

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

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

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

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

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

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

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

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

%Y Cf. A212829.

%K nonn

%O 1,2

%A _R. H. Hardin_, May 28 2012