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A243513 Number of length n+2 0..4 arrays with no three elements in a row with pattern aba or abb (with a!=b) and new values 0..4 introduced in 0..4 order.

%I A243513 #9 Nov 02 2018 12:27:31
%S A243513 3,5,10,24,65,187,552,1646,4927,14769,44294,132868,398589,1195751,
%T A243513 3587236,10761690,32285051,96855133,290565378,871696112,2615088313,
%U A243513 7845264915,23535794720,70607384134,211822152375,635466457097
%N A243513 Number of length n+2 0..4 arrays with no three elements in a row with pattern aba or abb (with a!=b) and new values 0..4 introduced in 0..4 order.
%H A243513 R. H. Hardin, <a href="/A243513/b243513.txt">Table of n, a(n) for n = 1..210</a>
%F A243513 Empirical: a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
%F A243513 Conjectures from _Colin Barker_, Nov 02 2018: (Start)
%F A243513 G.f.: x*(3 - 10*x + 6*x^2) / ((1 - x)^2*(1 - 3*x)).
%F A243513 a(n) = (7 + 3^n + 2*n) / 4.
%F A243513 (End)
%e A243513 All solutions for n=3:
%e A243513 ..0....0....0....0....0....0....0....0....0....0
%e A243513 ..1....0....1....0....0....1....1....1....0....0
%e A243513 ..2....0....2....1....0....2....2....2....0....1
%e A243513 ..3....1....0....2....0....0....3....3....0....2
%e A243513 ..0....2....1....3....1....3....1....4....0....0
%Y A243513 Column 4 of A243519.
%K A243513 nonn
%O A243513 1,1
%A A243513 _R. H. Hardin_, Jun 05 2014