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a(n) = number of n-lettered words in the alphabet {1, 2, 3} with as many occurrences of the substring (consecutive subword) [1, 1, 2] as of [1, 1, 3].
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%I #15 Mar 03 2017 19:39:17

%S 1,3,9,25,69,189,519,1431,3969,11077,31107,87867,249523,711987,

%T 2040201,5868009,16932645,49000221,142153251,413303355,1203986079,

%U 3513322887,10267781301,30048790725,88045917831,258268671963,758350570077,2228771296357,6555782946621,19298241524061,56848508815063

%N a(n) = number of n-lettered words in the alphabet {1, 2, 3} with as many occurrences of the substring (consecutive subword) [1, 1, 2] as of [1, 1, 3].

%H G. C. Greubel, <a href="/A211288/b211288.txt">Table of n, a(n) for n = 0..1000</a>

%H Shalosh B. Ekhad and Doron Zeilberger, <a href="http://arxiv.org/abs/1112.6207">Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type</a>, arXiv:1112.6207, 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence.

%F G.f.: 1/((1 - 2*x)*sqrt(1 - 2*x - 3*x^2)). - _G. C. Greubel_, Mar 03 2017

%p series(1/((1-2*x)*sqrt(1-2*x-3*x^2)),x=0,31); # _Mark van Hoeij_, May 10 2013

%t CoefficientList[Series[1/((1 - 2*x)*Sqrt[1 - 2*x - 3*x^2]), {x,0,50}], x] (* _G. C. Greubel_, Mar 03 2017 *)

%o (PARI) x='x+O('x^66); Vec(1/((1-2*x)*sqrt(1-2*x-3*x^2))) \\ _Joerg Arndt_, May 11 2013

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Apr 07 2012