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 A056343 #31 Oct 13 2018 09:34:38
%S A056343 0,0,1,6,18,56,147,411,1084,2979,8043,22244,61278,171030,477929,
%T A056343 1345236,3795750,10758902,30572427,87149124,248991822,713096352,
%U A056343 2046303339,5883433409,16944543810,48879769575
%N A056343 Number of bracelets of length n using exactly three different colored beads.
%C A056343 Turning over will not create a new bracelet.
%D A056343 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 A056343 Vincenzo Librandi, <a href="/A056343/b056343.txt">Table of n, a(n) for n = 1..1000</a>
%F A056343 a(n) = A027671(n) - 3*A000029(n) + 3.
%F A056343 From _Robert A. Russell_, Sep 26 2018: (Start)
%F A056343 a(n) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * Sum_{d|n} phi(d) * S2(n/d,k), where k=3 is the number of colors and S2 is the Stirling subset number A008277.
%F A056343 G.f.: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=3 is the number of colors. (End)
%e A056343 For a(4)=6, the arrangements are ABAC, ABCB, ACBC, AABC, ABBC, and ABCC. Only the last three are chiral, their reverses being AACB, ACBB, and ACCB respectively. - _Robert A. Russell_, Sep 26 2018
%t A056343 t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]);
%t A056343 T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
%t A056343 a[n_] := T[n, 3];
%t A056343 Array[a, 26] (* _Jean-François Alcover_, Nov 05 2017, after _Andrew Howroyd_ *)
%t A056343 k=3; Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#,k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n,1,30}] (* _Robert A. Russell_, Sep 26 2018 *)
%Y A056343 Column 3 of A273891.
%Y A056343 Equals (A056283 + A056489) / 2 = A056283 - A305542 = A305542 + A056489.
%K A056343 nonn
%O A056343 1,4
%A A056343 _Marks R. Nester_