A056303 Number of primitive (period n) n-bead necklace structures using exactly two different colored beads.

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

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