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%I #24 Apr 11 2024 01:39:58
%S 1,1,3,4,8,11,22,33,62,101,189,324,611,1087,2055,3770,7154,13363,
%T 25481,48174,92204,175791,337593,647325,1246862,2400841,4636389,
%U 8956059,17334800,33570815,65108061,126355335,245492243,477284181,928772649,1808538354,3524337979,6872209823
%N Number of necklaces with n beads colored white or red, where the number of white beads is odd and at least three and turning over is allowed.
%C 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.
%C 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
%H Andrew Howroyd, <a href="/A263768/b263768.txt">Table of n, a(n) for n = 3..100</a>
%H Bernd Mulansky and Andreas Potschka, <a href="https://arxiv.org/abs/2404.01841">A zonogon approach for computing small polygons of maximum perimeter</a>, arXiv:2404.01841 [math.OC], 2024. See p. 9.
%F a(n) = (A000016(n) + A016116(n-1)) / 2 - 1. - _Andrew Howroyd_, Feb 28 2017
%F a(n) = A007147(n) - 1. - _Bernd Mulansky_, Mar 08 2023
%e 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).
%t 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 *)
%Y Cf. A227910, A007147, A000016.
%K nonn,changed
%O 3,3
%A _David Eppstein_, Oct 25 2015
%E a(21)-a(40) from _Andrew Howroyd_, Feb 28 2017
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