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A263768 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. 1
1, 1, 3, 4, 8, 11, 22, 33, 62, 101, 189, 324, 611, 1087, 2055, 3770, 7154, 13363, 25481, 48174, 92204, 175791, 337593, 647325, 1246862, 2400841, 4636389, 8956059, 17334800, 33570815, 65108061, 126355335, 245492243, 477284181, 928772649, 1808538354, 3524337979, 6872209823 (list; graph; refs; listen; history; text; internal format)



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


Andrew Howroyd, Table of n, a(n) for n = 3..100


a(n) = (A000016(n) + A016116(n-1)) / 2 - 1. - Andrew Howroyd, Feb 28 2017


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).


Cf. A227910, A007147, A000016.

Sequence in context: A109794 A034417 A126873 * A050316 A161538 A319980

Adjacent sequences:  A263765 A263766 A263767 * A263769 A263770 A263771




David Eppstein, Oct 25 2015


a(21)-a(40) from Andrew Howroyd, Feb 28 2017



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Last modified September 21 19:59 EDT 2019. Contains 327282 sequences. (Running on oeis4.)