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%I #30 Feb 28 2018 18:35:09
%S 0,1,2,5,9,19,35,71,135,271,527,1055,2079,4159,8255,16511,32895,65791,
%T 131327,262655,524799,1049599,2098175,4196351,8390655,16781311,
%U 33558527,67117055,134225919,268451839,536887295,1073774591,2147516415,4295032831,8590000127
%N Number of reversible string structures with n beads using exactly two different colors.
%C A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.
%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 Alois P. Heinz, <a href="/A056326/b056326.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3, 0, -6, 4).
%F a(n) = A005418(n) - 1.
%F From _Colin Barker_, Nov 25 2012: (Start)
%F a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
%F G.f.: x^2*(x^2+x-1)/((x-1)*(2*x-1)*(2*x^2-1)). (End)
%t Table[(StirlingS2[n,2] + StirlingS2[Floor[n/2]+1,2])/2, {n,1,30}] (* _Robert A. Russell_, Jan 29 2018 *)
%t LinearRecurrence[{3, 0, -6, 4}, {0, 1, 2, 5}, 35] (* or *)
%t Rest@ CoefficientList[Series[x^2*(x^2 + x - 1)/((x - 1) (2 x - 1) (2 x^2 - 1)), {x, 0, 35}], x] (* _Michael De Vlieger_, Jan 31 2018 *)
%Y Column k=2 of A284949 and of A291883.
%Y Cf. A056309.
%K nonn,easy
%O 1,3
%A _Marks R. Nester_
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