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%I #48 Mar 09 2024 12:32:20
%S 0,1,1,3,3,7,9,19,29,55,93,179,315,595,1095,2067,3855,7315,13797,
%T 26271,49939,95419,182361,349715,671091,1290871,2485533,4794087,
%U 9256395,17896831,34636833,67110931,130150587,252648991,490853415,954444607,1857283155,3616828363
%N Number of n-bead necklace structures using exactly two different colored beads.
%C Turning over the necklace is not allowed. Colors may be permuted without changing the necklace 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 Vincenzo Librandi, <a href="/A056295/b056295.txt">Table of n, a(n) for n = 1..1000</a>
%H Joshua P. Bowman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Bowman/bowman4.html">Compositions with an Odd Number of Parts, and Other Congruences</a>, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.
%F a(n) = A000013(n) - 1.
%F From _Robert A. Russell_, Mar 08 2018: (Start)
%F G.f.: Sum_{ d>0 } phi(d)*(2*log(1-x^d) - (1+[d == 0 mod 2])*log(1-2*x^d)) / (2*d);
%F a(n) = (1/n)*Sum_{d|n} phi(d) * S2(n/d + [d == 0 mod 2], 2), where S2(n, k) is the Stirling subset number, A008277. (End)
%e For a(7) = 9, the color patterns are AAAAAAB, AAAAABB, AAAABAB, AAAABBB, AAABAAB, AABAABB, AABABAB, AAABABB, and AAABBAB. The first seven are achiral; the last two are a chiral pair. - _Robert A. Russell_, Mar 08 2018
%p See A000013.
%t Table[DivisorSum[n, EulerPhi[#] If[OddQ[#], StirlingS2[n/#, 2], StirlingS2[n/#+1, 2]]&]/n, {n,1,30}] (* _Robert A. Russell_, Feb 20 2018 *)
%Y Column 2 of A152175.
%Y Cf. A000013, A052823.
%K nonn,easy
%O 1,4
%A _Marks R. Nester_
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