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

Author

Raphie Frank, Sep 30 2012

Keywords

• sign
• fini
• full

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).

Examples

			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.
		

Links

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