A227910 -id:A227910 - cp's OEIS frontend

A262232 Number of black and white n-bead necklaces with at least 3 white beads (turning over is not allowed); also number of clockwise arrangements of reflex and non-reflex angles that can form an n-gon.

0, 0, 0, 1, 2, 4, 9, 15, 30, 54, 101, 181, 344, 624, 1173, 2183, 4106, 7702, 14591, 27585, 52476, 99868, 190733, 364711, 699238, 1342170, 2581413, 4971053, 9587564, 18512776, 35792551, 69273651, 134219778, 260301158

Offset: 0

Danny Rorabaugh, Sep 15 2015

nonn

A reflex angle is an angle with measure greater than Pi or 180 degrees. Every polygon has at least three angles with measure less than Pi or 180 degrees.

			Let 1's represent black beads and 0's represent white beads. For n=6, the a(6)=9 necklaces are 000000, 000001, 000011, 000101, 000111, 001001, 001011, 001101, 010101. Note that 001011 and 001101 would be equivalent if "turning over" were allowed.

Danny Rorabaugh, Table of n, a(n) for n = 0..500
Danny Rorabaugh, Polygon demonstration of a(6)=9

Cf. A000031, A227910.

[sum([Necklaces([n-k,k]).cardinality()  for k in range(n-2)]) for n in range(34)]

a(n) = A000031(n) - 2 - floor(n/2), n>0.

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.

Original entry on oeis.org

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

Offset: 3

David Eppstein, Oct 25 2015

nonn

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

			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).

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. See also Math. Program., 2025.

Cf. A227910, A007147, A000016.

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 *)

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

a(n) = A007147(n) - 1. - Bernd Mulansky, Mar 08 2023

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

A298612 The number of concave polygon classes.

Original entry on oeis.org

0, 1, 3, 8, 14, 29, 53, 100, 180, 343, 623, 1172, 2182, 4105, 7701, 14590, 27584, 52475, 99867, 190732, 364710, 699237, 1342169, 2581412, 4971052, 9587563, 18512775, 35792550, 69273650, 134219777, 260301157, 505294108, 981706812

Offset: 3

Stuart E Anderson, Jan 23 2018

nonn

A concave polygon has at least one concave interior corner angle, and at least three convex interior corner angles. Two concave polygon classes are equivalent if the cyclic ordering of the concave and convex interior angles of each are equal.

a(n) is also the number of combinatorial necklaces with n beads in 2 colors (black and white) with at least one white bead and no fewer than 3 black beads.

Jason Davies, Combinatorial necklaces and bracelets
Peter Kagey, Illustrations of a(4), a(5), a(6)

Cf. A000031, A004526, A227910, A262232.

Table[DivisorSum[n, EulerPhi[#] 2^(n/#) &]/n - Floor[n/2] - 3, {n, 3, 35}] (* Michael De Vlieger, Jan 28 2018 *)

a(n) = A000031(n) - A004526(n) - 3, n >= 3.

a(n) = A262232(n)-1, n >= 3.

cp's OEIS Frontend

A262232 Number of black and white n-bead necklaces with at least 3 white beads (turning over is not allowed); also number of clockwise arrangements of reflex and non-reflex angles that can form an n-gon.

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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.

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A298612 The number of concave polygon classes.

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