A298612 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,3

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

Table of n, a(n) for n=3..35.

Jason Davies, Combinatorial necklaces and bracelets

Peter Kagey, Illustrations of a(4), a(5), a(6)

FORMULA

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

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

MATHEMATICA

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

CROSSREFS

Cf. A000031, A004526, A227910, A262232.

Sequence in context: A268191 A169929 A129067 * A340627 A350520 A168155