cp's OEIS Frontend

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

%I A056303 #23 Aug 22 2017 20:53:08
%S A056303 0,1,1,2,3,5,9,16,28,51,93,170,315,585,1091,2048,3855,7280,13797,
%T A056303 26214,49929,95325,182361,349520,671088,1290555,2485504,4793490,
%U A056303 9256395,17895679,34636833,67108864,130150493,252645135,490853403,954437120,1857283155
%N A056303 Number of primitive (period n) n-bead necklace structures using exactly two different colored beads.
%C A056303 Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.
%C A056303 Identical to A000048 for n>1.
%C A056303 Number of binary Lyndon words of length n with an odd number of zeros. - _Joerg Arndt_, Oct 26 2015
%D A056303 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]
%F A056303 a(n) = Sum mu(d)*A056295(n/d) where d divides n.
%F A056303 a(n) =  A000048(n) - A000007(n-1).
%o A056303 (PARI) vector(100, n, sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d)))/(2*n)-!(n-1)) \\ _Altug Alkan_, Oct 26 2015
%o A056303 (Python)
%o A056303 from sympy import divisors, mobius
%o A056303 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)
%o A056303 def a(n): return a000048(n) - 0**(n - 1) # _Indranil Ghosh_, Apr 28 2017
%Y A056303 Column 2 of A107424.
%Y A056303 Cf. A000007, A000048, A001037, A056295.
%K A056303 nonn
%O A056303 1,4
%A A056303 _Marks R. Nester_