This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A056309 #53 Sep 08 2022 08:45:01
%S A056309 0,1,4,8,18,34,70,134,270,526,1054,2078,4158,8254,16510,32894,65790,
%T A056309 131326,262654,524798,1049598,2098174,4196350,8390654,16781310,
%U A056309 33558526,67117054,134225918,268451838,536887294,1073774590,2147516414,4295032830,8590000126
%N A056309 Number of reversible strings with n beads using exactly two different colors.
%C A056309 A string and its reverse are considered to be equivalent.
%D A056309 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 A056309 Harvey P. Dale, <a href="/A056309/b056309.txt">Table of n, a(n) for n = 1..1000</a>
%H A056309 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-6,4).
%F A056309 a(n) = A005418(n+1) - 2.
%F A056309 G.f.: x^2*(1 + x - 4*x^2)/(1 - 3*x + 6*x^3 - 4*x^4). - _Colin Barker_, Feb 03 2012
%F A056309 a(1)=0, a(2)=1, a(3)=4, a(4)=8, a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4). - _Harvey P. Dale_, Feb 18 2012
%F A056309 From _Colin Barker_, Nov 24 2017: (Start)
%F A056309 a(n) = 2^(n/2-1) + 2^(n-1) - 2 for n even.
%F A056309 a(n) = 2^((n-1)/2) + 2^(n-1) - 2 for n odd. (End)
%F A056309 a(n) = A000079(n-1) + A056453(n-2). - _Peter Luschny_, Nov 25 2017
%F A056309 a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=2 is the number of colors and S2 is the Stirling subset number. - _Robert A. Russell_, Sep 25 2018
%e A056309 For n=3, the four rows are ABA, BAB, AAB, and ABB, the last two being respectively equivalent to BAA and BBA, with which they form chiral pairs. - _Robert A. Russell_, Sep 25 2018
%p A056309 seq(2^(n-1) + 2^floor((n-1)/2) - 2, n=1..34); # _Peter Luschny_, Nov 25 2017
%t A056309 Rest[CoefficientList[Series[x^2(1+x-4x^2)/(1-3x+6x^3-4x^4),{x,0,30}],x]] (* or *) LinearRecurrence[{3,0,-6,4},{0,1,4,8},30] (* _Harvey P. Dale_, Feb 18 2012 *)
%o A056309 (PARI) Vec(x^2*(1+x-4*x^2)/(1-3*x+6*x^3-4*x^4) + O(x^40)) \\ _Colin Barker_, Nov 24 2017
%o A056309 (PARI) a(n) = 2^(n-1)+2^((n-1)\2)-2; \\ _Altug Alkan_, Sep 25 2018
%o A056309 (Magma) [2^(n-1)+2^((n-1) div 2)-2: n in [1..40]]; // _Vincenzo Librandi_, Sep 29 2018
%Y A056309 Cf. A000079, A005418.
%Y A056309 Equals (A000918 + A056453) / 2.
%Y A056309 a(n) = A000918(n) - A122746(n-2) = A122746(n-2) + A056453(n).
%K A056309 nonn,easy
%O A056309 1,3
%A A056309 _Marks R. Nester_