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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

(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

Paolo Xausa, Table of n, a(n) for n = 1..3000

FORMULA

a(n) = Sum mu(d)*A056295(n/d) where d divides n.

a(n) = A000048(n) - A000007(n-1).

MATHEMATICA

A056303[n_] := If[n == 1, 0, DivisorSum[n, MoebiusMu[#]*2^(n/#) &, OddQ]/(2*n)];

Array[A056303, 50] (* Paolo Xausa, Nov 26 2025 *)

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