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 A056295 Number of n-bead necklace structures using exactly two different colored beads. 9
 0, 1, 1, 3, 3, 7, 9, 19, 29, 55, 93, 179, 315, 595, 1095, 2067, 3855, 7315, 13797, 26271, 49939, 95419, 182361, 349715, 671091, 1290871, 2485533, 4794087, 9256395, 17896831, 34636833, 67110931, 130150587, 252648991, 490853415, 954444607, 1857283155, 3616828363 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure. REFERENCES 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.] LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 FORMULA a(n) = A000013(n) - 1. From Robert A. Russell, Mar 08 2018: (Start) G.f.: Sum_{ d>0 } phi(d)*(2*log(1-x^d) - (1+[d == 0 mod 2])*log(1-2*x^d)) / (2*d); 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) EXAMPLE 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 MAPLE See A000013. MATHEMATICA Table[DivisorSum[n, EulerPhi[#] If[OddQ[#], StirlingS2[n/#, 2], StirlingS2[n/#+1, 2]]&]/n, {n, 1, 30}] (* Robert A. Russell, Feb 20 2018 *) CROSSREFS Column 2 of A152175. Cf. A000013, A052823. Sequence in context: A122012 A185306 A320314 * A117525 A075149 A161618 Adjacent sequences:  A056292 A056293 A056294 * A056296 A056297 A056298 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified April 4 08:58 EDT 2020. Contains 333213 sequences. (Running on oeis4.)