OFFSET
3,3
COMMENTS
a(n) is also the number of non-isomorphic n-vertex undirected graphs forming an odd cycle with any number of degree-1 vertices attached to each cycle vertex. To transform a necklace into a graph of this type, create a cycle vertex for each white bead and a pendant vertex for each red bead, with each pendant vertex attached to the next clockwise cycle vertex. Since these are exactly the graphs of the n-vertex and n-edge linear thrackles, a(n) is also the number of non-isomorphic linear thrackles.
For any n there is a unique n-bead necklace where the number of white beads is 1. Hence this sequence is one less than the number of n-bead (0,1) bracelets with an odd number of 0's. - Andrew Howroyd, Feb 28 2017
LINKS
Andrew Howroyd, Table of n, a(n) for n = 3..100
Bernd Mulansky and Andreas Potschka, A zonogon approach for computing small polygons of maximum perimeter, arXiv:2404.01841 [math.OC], 2024. See p. 9.
FORMULA
a(n) = A007147(n) - 1. - Bernd Mulansky, Mar 08 2023
EXAMPLE
For n=5 the a(5)=3 solutions are: five white beads (a 5-cycle), three white beads and two red beads with the two red beads adjacent (a triangle with two pendant vertices attached at one triangle vertex), and three white beads and two red beads with the two red beads separated (a triangle with two of its vertices having a single pendant vertex attached).
MATHEMATICA
Table[1/2*(2^Quotient[n - 1, 2] + Total@ Map[(Mod[#, 2]*EulerPhi[#]*2^(n/#) &), Divisors[n]]/(2 n)) - 1, {n, 3, 40}] (* Michael De Vlieger, Apr 10 2024, after Jean-François Alcover at A007147 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David Eppstein, Oct 25 2015
EXTENSIONS
a(21)-a(40) from Andrew Howroyd, Feb 28 2017
STATUS
approved