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A268255 Number of length-(n+1) 0..2 arrays with new repeated values introduced in sequential order starting with zero.

%I A268255 #8 Jan 11 2019 15:14:17
%S A268255 7,17,42,106,273,717,1918,5218,14413,40349,114282,326938,943257,
%T A268255 2740797,8010982,23529346,69385813,205282157,608959218,1810358938,
%U A268255 5391414273,16078923309,48007516942,143470822498,429083952157,1284051486077
%N A268255 Number of length-(n+1) 0..2 arrays with new repeated values introduced in sequential order starting with zero.
%H A268255 R. H. Hardin, <a href="/A268255/b268255.txt">Table of n, a(n) for n = 1..210</a>
%F A268255 Empirical: a(n) = 7*a(n-1) - 15*a(n-2) + 7*a(n-3) + 6*a(n-4).
%F A268255 Conjectures from _Colin Barker_, Jan 11 2019: (Start)
%F A268255 G.f.: x*(7 - 32*x + 28*x^2 + 18*x^3) / ((1 - 2*x)*(1 - 3*x)*(1 - 2*x - x^2)).
%F A268255 a(n) = 2^n + 3^n/2 + (3/4-1/sqrt(2))*(1-sqrt(2))^n + (3/4+1/sqrt(2))*(1+sqrt(2))^n.
%F A268255 (End)
%e A268255 Some solutions for n=8:
%e A268255 ..0....1....1....0....0....1....0....0....2....1....0....1....2....0....0....1
%e A268255 ..0....0....0....0....0....0....0....2....0....2....1....2....1....1....0....0
%e A268255 ..2....0....0....2....0....1....1....0....0....0....2....1....2....0....1....2
%e A268255 ..0....1....0....0....2....0....0....0....0....0....1....2....1....0....0....0
%e A268255 ..1....1....1....1....0....2....0....0....1....2....2....0....0....2....0....2
%e A268255 ..0....1....2....1....2....0....1....1....1....1....1....0....1....1....1....1
%e A268255 ..1....1....1....0....0....0....0....0....1....0....2....1....0....0....1....0
%e A268255 ..0....1....2....1....2....2....2....0....2....1....0....0....0....1....1....0
%e A268255 ..2....1....0....0....1....1....1....1....2....0....0....2....1....1....1....1
%Y A268255 Column 2 of A268261.
%K A268255 nonn
%O A268255 1,1
%A A268255 _R. H. Hardin_, Jan 29 2016