A298612 The number of concave polygon classes.

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

Author

Stuart E Anderson, Jan 23 2018

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

Cf. A000031, A004526, A227910, A262232.

Programs

Mathematica

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

Formula

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

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