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

Author

Danny Rorabaugh, Sep 15 2015

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000031, A227910.

Programs

Sage

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

Formula

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