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 A056308 #56 Aug 15 2024 07:05:29
%S A056308 1,6,21,126,666,3996,23436,140616,840456,5042736,30236976,181421856,
%T A056308 1088414496,6530486976,39182222016,235093332096,1410555793536,
%U A056308 8463334761216,50779983373056,304679900238336,1828079250264576,10968475501587456,65810852102532096
%N A056308 Number of reversible strings with n beads using a maximum of six different colors.
%C A056308 A string and its reverse are considered to be equivalent. Thus aabc and cbaa are considered to be identical, but abca is a different string.
%D A056308 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 A056308 Colin Barker, <a href="/A056308/b056308.txt">Table of n, a(n) for n = 0..1000</a>
%H A056308 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,6,-36).
%F A056308 a(n) = (6^n + 6^floor((n+1)/2))/2.
%F A056308 G.f.: (1-21*x^2) / ((1-6*x)*(1-6*x^2)). - _R. J. Mathar_, Jul 06 2011 [Adapted to offset 0 by _Robert A. Russell_, Nov 10 2018]
%F A056308 a(n) = 6*a(n-1) + 6*a(n-2) - 36*a(n-3) for n > 3. - _Colin Barker_, Mar 20 2017
%F A056308 a(n) = (A000400(n) + A056452(n)) / 2. - _Robert A. Russell_, Jun 19 2018
%F A056308 a(n) = 6^(n + floor((n-1)/2)) * a(1-n) for all n in Z. - _Michael Somos_, Jul 10 2018
%e A056308 For a(2)=21, the six achiral strings are AA, BB, CC, DD, EE, and FF; the 15 (equivalent) chiral pairs are AB-BA, AC-CA, AD-DA, AE-EA, AF-FA, BC-CB, BD-DB, BE-EB, BF-FB, CD-DC, CE-EC, CF-FC, DE-ED, DF-FD, and EF-FE.
%t A056308 k=6; Table[(k^n+k^Ceiling[n/2])/2,{n,0,30}] (* _Robert A. Russell_, Nov 25 2017 *)
%t A056308 a[ n_] := (6^n + 6^Quotient[n + 1, 2]) / 2; (* _Michael Somos_, Jul 10 2018 *)
%t A056308 LinearRecurrence[{6, 6, -36}, {1, 6, 21}, 31] (* _Robert A. Russell_, Nov 10 2018 *)
%o A056308 (PARI) Vec((1-21*x^2) / ((1 - 6*x)*(1 - 6*x^2)) + O(x^30)) \\ _Colin Barker_, Mar 20 2017 [Adapted to offset 0 by _Robert A. Russell_, Nov 10 2018]
%o A056308 (PARI) {a(n) = (6^n + 6^((n+1)\2)) / 2}; \\ _Michael Somos_, Jul 10 2018
%o A056308 (Magma) I:=[1,6,21]; [n le 3 select I[n] else 6*Self(n-1) +6*Self(n-2) - 36*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Nov 10 2018
%Y A056308 Column 6 of A277504.
%Y A056308 Cf. A000400 (oriented), A320524 (chiral), A056452 (achiral).
%K A056308 nonn,easy
%O A056308 0,2
%A A056308 _Marks R. Nester_
%E A056308 a(0)=1 prepended by _Robert A. Russell_, Nov 10 2018