A217290 Integers n such that 2*cos(2*Pi/n) is an integer.

-6, -4, -3, -2, -1, 1, 2, 3, 4, 6

Offset

0,1

Comments

Terms are the allowable n-fold rotational symmetries of a crystal (rotation by 360 degrees/n leaves the object unchanged).

The positive values of this sequence {1, 2, 3, 4, 6} are the proper divisors of 12, all having a totient of 1 or 2 (see A000010).

Links

Table of n, a(n) for n=0..9.

W. Scherrer, Die Einlagerung eines regulären Vielecks in ein Gitter, Elemente der Mathematik, 1946, 1(6), p.97-98.

Wikipedia, Crystallographic Restriction Theorem

Example

2*cos(2Pi/1) =  2
2*cos(2Pi/2) = -2
2*cos(2Pi/3) = -1
2*cos(2Pi/4) =  0
2*cos(2Pi/6) =  1
2*cos(2Pi/10) = 1.6180339887... and so 10, for instance, is not in this sequence.

Crossrefs

Sequence in context: A011408 A350094 A197760 * A157296 A329081 A343461

Adjacent sequences: A217287 A217288 A217289 * A217291 A217292 A217293

Keyword

sign,fini,full

Author

Raphie Frank, Sep 30 2012

Status

approved