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 A278642 #26 Oct 13 2018 09:32:17
%S A278642 0,0,0,20,105,672,3535,19350,102795,556010,3010098,16467450,90619690,
%T A278642 502194420,2798240265,15671993560,88156797855,497837886000,
%U A278642 2821092554035,16035752398770,91403856697944,522308167195260,2991401733402075,17168047238861070,98716274117752900,568605754068247644,3280417827002225910,18953525314104758810
%N A278642 Number of pairs of orientable necklaces with n beads and up to 6 colors; i.e., turning the necklace over does not leave it unchanged. The turned-over necklace is not included in the count.
%C A278642 Number of chiral bracelets of n beads using up to six different colors.
%F A278642 Equals (A054625(n) - A056488(n)) / 2 = A054625(n) - A056341(n) = A056341(n) - A056488(n), for n >= 1.
%F A278642 G.f.: k = 6, (1 - Sum_{n >= 1} phi(n)*log(1 - k*x^n)/n - Sum_{i = 0..2} Binomial[k, i]*x^i / ( 1 - k*x^2) )/2.
%F A278642 For n > 0, a(n) = -(k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/2n)* Sum_{d|n} phi(d)*k^(n/d), where k = 6 is the maximum number of colors. - _Robert A. Russell_, Sep 24 2018
%t A278642 mx = 40; f[x_, k_] := (1 - Sum[EulerPhi[n] * Log[1 - k * x^n]/n,{n, mx}] - Sum[Binomial[k, i] * x^i, {i, 0, 2}]/(1 - k * x^2))/2; CoefficientList[Series[f[x, 6], {x, 0, mx}], x]
%t A278642 k = 6; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) - (k^Floor[(n + 1)/2] + k^Ceiling[(n + 1)/2])/4, {n, 30}], 0] (* _Robert A. Russell_, Sep 24 2018 *)
%Y A278642 Column 6 of A293496.
%Y A278642 Cf. A059076 (2 colors), A278639 (3 colors), A278640 (4 colors), A278641 (5 colors).
%K A278642 nonn
%O A278642 0,4
%A A278642 _Herbert Kociemba_, Nov 24 2016