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%I #14 Mar 03 2017 19:44:38
%S 1,2,2,2,4,6,12,20,40,70,140,252,504,924,1848,3432,6864,12870,25740,
%T 48620,97240,184756,369512,705432,1410864,2704156,5408312,10400600,
%U 20801200,40116600,80233200,155117520,310235040,601080390,1202160780,2333606220,4667212440,9075135300,18150270600,35345263800
%N a(n) = number of n-lettered words in the alphabet {1, 2} with as many occurrences of the substring (consecutive subword) [1, 1] as of [2, 2].
%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 preprint arXiv:1112.6207, 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence. [From _N. J. A. Sloane_, Apr 07 2012]
%F G.f.: 1 + x + x*sqrt((1+2*x)/(1-2*x))= 1 + x + x/G(0), where G(k)= 1 - 2*x/(1 + 2*x/(1 + 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 26 2013
%Y Apart from initial terms, same as A063886.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Apr 07 2012
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