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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

Table of n, a(n) for n=1..37.

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

Marks R. Nester

STATUS

approved

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Last modified October 15 00:14 EDT 2019. Contains 328025 sequences. (Running on oeis4.)