This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A186314 #20 Apr 08 2024 06:56:03
%S A186314 0,0,1,6,26,99,352,1200,3977,12918,41338,130779,410048,1276512,
%T A186314 3950929,12170598,37343834,114209811,348332320,1059927312,3218870105,
%U A186314 9758944470,29544747706,89335651851,269843267456,814337329344,2455598257057,7399746051270
%N A186314 Number of ternary strings of length n which contain 01.
%H A186314 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,3).
%F A186314 a(n) = 3*a(n-1) + (3^(n-2) - a(n-2)).
%F A186314 G.f.: x^2/((1-3*x)*(1-3*x+x^2)). a(n) = 3^n - A001906(n+1). - _Bruno Berselli_, Feb 23 2011
%e A186314 The recursive formula is based on extending such a string of length n-1 with {0,1,2} or extending a non-matching string of length (n-2) with "01". For n=2, there is just 1 string: "01". For n=3, we append {0,1,2} to "01" and append "01" to {"0","1","2"}, the three non-matching strings of length 1, for a total of a(3)=6.
%t A186314 nn=20;CoefficientList[Series[1/(1-3x)-1/(x^2+(1-3x)),{x,0,nn}],x] (* _Geoffrey Critzer_, Dec 25 2013 *)
%t A186314 LinearRecurrence[{6,-10,3},{0,0,1},30] (* _Harvey P. Dale_, Jun 14 2020 *)
%Y A186314 Cf. A186244 (ternary strings which contain 00).
%K A186314 nonn,easy
%O A186314 0,4
%A A186314 _Toby Gottfried_, Feb 17 2011