A262232 -id:A262232 - cp's OEIS frontend

A227910 The number of necklaces with n beads of white and red colors, including at least three white ones.

1, 2, 4, 8, 13, 24, 40, 71, 119, 216, 372, 678, 1215, 2240, 4102, 7674, 14299, 27000, 50952, 96896, 184397, 352684, 675174, 1296843, 2493711, 4806062, 9272764, 17920843, 34669585, 67159032, 130216106, 252745349, 490984469, 954637538, 1857545280, 3617214660, 7048675939, 13744694906

Offset: 3

Vladimir Letsko, Oct 13 2013

nonn

a(n) is the number of classes (not necessarily convex) n-gons. Two n-gons are assigned to the same class, if (under suitable start and direction) their vertex at the same time belong or do not belong to the convex hull of the corresponding polygon.
a(n) is also the number of classes (not necessarily convex) n-gons at the other classification. Two n-gons belong to one class, if (under suitable start and direction) their angles simultaneously have a measure either less or greater than 180 degrees.
Note that the equation quantities of classes of these classifications does not mean equality classes themselves.
a(n) is also the number of non-isomorphic n-vertex undirected graphs in the form of a simple 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 the pendant vertex for a red bead attached to the cycle vertex for the clockwise next white bead. - David Eppstein, Oct 25 2015

			a(4)=2 because there are exactly 2 necklaces with 4 beads of white and red colors, including at least three white ones:
1) all beads are white;
2) one bead is red.

Andrew Howroyd, Table of n, a(n) for n = 3..200
Peter Kagey, Examples of classes of n-gons for a(4), a(5), and a(6)
Vladimir Letsko, Mathematical Marathon, Problem 150 (in Russian)

Cf. A000029, A262232.

g:=(n,m)->add(phi(d)*binomial(n/d,m/d),d in divisors(igcd(n,m)))/2/n+1/2*binomial(floor(m/2)+floor((n-m)/2),floor(m/2));
for n from 3 do s:=0:for m from 3 to n dos:=s+g(n,m) od: print(n,s) od:

g[n_, m_] := Sum[EulerPhi[d]*Binomial[n/d, m/d], {d , Divisors[GCD[n, m]]}]/2/n + 1/2*Binomial[Floor[m/2] + Floor[(n - m)/2], Floor[m/2]];
A227910 = Table[Sum[g[n, m], {m, 3, n}], {n, 3, 40}] (* Jean-François Alcover, Jul 02 2018, from Maple *)

a(n) = (sum(m=3, n, (1/n*sumdiv(gcd(m,n), d, eulerphi(d)*binomial(n/d,m/d))) + binomial(m\2+(n-m)\2, m\2) ))/2; \\ Andrew Howroyd, Oct 11 2017

a(n) = (Sum_{m=3..n} (1/n*sum_{d|gcd(m,n)} phi(d)*binomial(n/d,m/d)) + binomial(floor(m/2)+floor((n-m)/2), floor(m/2)))/2.

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

A227910 The number of necklaces with n beads of white and red colors, including at least three white ones.

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