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%I #21 Oct 28 2016 10:25:01
%S 2,5,16,38,96,220,512,1144,2560,5616,12288,26592,57344,122816,262144,
%T 556928,1179648,2490112,5242880,11009536,23068672,48233472,100663296,
%U 209713152,436207616,905965568,1879048192,3892305920,8053063680,16642981888,34359738368
%N Number of ternary Lyndon words of length n with exactly two 1's.
%C If the offsets are modified, A124720 to A124723 are the 2nd to 5th Witt transform of A000079 [Moree]. - _R. J. Mathar_, Nov 08 2008
%C a(n+2) is the number of distinct unordered pairs of binay words having a total length of n letters: a(2+2) = 5 because we have the unordered pairs: (e,00),(e,01), (e,10), (e,11), (0,1) where e represents the empty word. Each pair has a total of 2 letters and the two elements of each pair are distinct words. - _Geoffrey Critzer_, Feb 28 2013
%H Colin Barker, <a href="/A124720/b124720.txt">Table of n, a(n) for n = 3..1000</a>
%H Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From _R. J. Mathar_, Nov 08 2008]
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-8,8).
%F G.f.: x^3*(2-3 x)/((1-2 x^2)(1- 2x)^2) = (x^2/(1-2x)^2 - x^2/(1-2*x^2))/2.
%F From _Colin Barker_, Oct 28 2016: (Start)
%F a(n) = 2^(n-3)*(n-1)-2^(n/2-2) for n even.
%F a(n) = 2^(n-3)*n-2^(n-3) for n odd.
%F a(n) = 4*a(n-1)-2*a(n-2)-8*a(n-3)+8*a(n-4) for n>6.
%F (End)
%e a(4) = 5 because 1122, 1123, 1132, 1213, 1133 are all Lyndon words on 3 letters with 2 ones.
%t nn=30;Drop[CoefficientList[Series[(1/(1-2x)^2-1/(1-2x^2))/2,{x,0,nn}],x],1] (* _Geoffrey Critzer_, Feb 28 2013 *)
%o (PARI) Vec(x^3*(2-3*x)/((1-2*x)^2*(1-2*x^2)) + O(x^40)) \\ _Colin Barker_, Oct 28 2016
%Y Cf. A051168, A027376, A124721, A124722, A124723.
%K nonn,easy
%O 3,1
%A _Mike Zabrocki_, Nov 05 2006
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