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 A056303 Number of primitive (period n) n-bead necklace structures using exactly two different colored beads. 7
 0, 1, 1, 2, 3, 5, 9, 16, 28, 51, 93, 170, 315, 585, 1091, 2048, 3855, 7280, 13797, 26214, 49929, 95325, 182361, 349520, 671088, 1290555, 2485504, 4793490, 9256395, 17895679, 34636833, 67108864, 130150493, 252645135, 490853403, 954437120, 1857283155 (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. Identical to A000048 for n>1. Number of binary Lyndon words of length n with an odd number of zeros. - Joerg Arndt, Oct 26 2015 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 FORMULA a(n) = Sum mu(d)*A056295(n/d) where d divides n. a(n) =  A000048(n) - A000007(n-1). PROG (PARI) vector(100, n, sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d)))/(2*n)-!(n-1)) \\ Altug Alkan, Oct 26 2015 (Python) from sympy import divisors, mobius def a000048(n): return 1 if n<1 else sum([mobius(d)*2**(n/d) for d in divisors(n) if d%2 == 1])/(2*n) def a(n): return a000048(n) - 0**(n - 1) # Indranil Ghosh, Apr 28 2017 CROSSREFS Column 2 of A107424. Cf. A000007, A000048, A001037, A056295. Sequence in context: A143961 A128023 A000048 * A074099 A006788 A054650 Adjacent sequences:  A056300 A056301 A056302 * A056304 A056305 A056306 KEYWORD nonn AUTHOR STATUS approved

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Last modified September 27 10:28 EDT 2021. Contains 347689 sequences. (Running on oeis4.)