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

0,5

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.

Links

Danny Rorabaugh, Table of n, a(n) for n = 0..500

Danny Rorabaugh, Polygon demonstration of a(6)=9

Formula

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

Example

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.

Prog

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

Crossrefs

Cf. A000031, A227910.

Sequence in context: A330737 A083270 A000879 * A358830 A321410 A218912

Adjacent sequences: A262229 A262230 A262231 * A262233 A262234 A262235

Keyword

nonn

Author

Danny Rorabaugh, Sep 15 2015

Status

approved