%I #31 Sep 28 2018 10:05:15
%S 0,0,3,18,78,273,921,2916,9150,28065,85773,259848,785778,2367813,
%T 7128201,21427956,64382550,193326105,580372293,1741847328,5227116378,
%U 15684323853,47059266081,141189861996
%N Number of reversible strings with n beads using exactly three different colors.
%C A string and its reverse are considered to be equivalent.
%D M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
%H Vincenzo Librandi, <a href="/A056310/b056310.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6,-24,49,6,-66,36).
%F a(n) = A032120(n) - 3*A005418(n+1) + 3.
%F G.f.: -3*x^3*(12*x^4 - 5*x^3 - 4*x^2 + 1)/((x - 1)*(2*x - 1)*(3*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)). [_Colin Barker_, Jul 07 2012]
%e For n=3, the three rows are ABC, ACB, and BAC, being respectively equivalent to CBA, BCA, and CAB, with which they form chiral pairs. - _Robert A. Russell_, Sep 25 2018
%p seq(coeff(series(-3*x^3*(12*x^4-5*x^3-4*x^2+1)/((x-1)*(2*x-1)*(3*x-1)*(2*x^2-1)*(3*x^2-1)),x,n+1), x, n), n = 1..25); # _Muniru A Asiru_, Sep 27 2018
%t k=3; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,k,30}] (* _Robert A. Russell_, Nov 25 2017 *)
%t LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 3, 18, 78, 273, 921}, 40] (* _Vincenzo Librandi_, Sep 27 2018 *)
%Y Cf. A032120, A005418.
%Y Column 3 of A305621.
%Y Equals (A001117 + A056454) / 2 = A001117 - A305623 = A305623 + A056454.
%K nonn,easy
%O 1,3
%A _Marks R. Nester_