A098913 Number of different ways angles from Pi/n to (n-1)Pi/n can tile around a vertex, where rotations of an angle sequence are not counted, but reflections that are different are counted.

1, 5, 19, 75, 287, 1053, 3859, 14089, 51463, 188697, 695155, 2573235, 9571195, 35759799, 134154259, 505163055, 1908619755, 7233118641, 27486768415, 104713346699, 399818311219, 1529747101965, 5864045590035, 22517965253595, 86607619323751, 333599840675337

Offset: 2

Author

Stuart E Anderson, Oct 17 2004

Keywords

• nonn

Comments

The sequence represents the number of ways rhombi (with appropriate angles) can tile around a vertex, e.g. a(5) is the number of ways Penrose rhombs can tile a vertex where tilings that are different by rotation are counted and tilings that are the same by reflection are also counted.

Also, the number of nonequivalent compositions of 2*n with maximum part size n-1 up to rotation. - Andrew Howroyd, Sep 06 2017

Examples

			a(4)=19 because 2pi = 3'3'2' or 2'2'2'2' or 3'1'2'2' or 3'1'3'1' or 3'2'1'2' or 3'2'2'1' or 3'3'1'1' or 2'2'1'2'1' or 2'2'2'1'1' or 3'1'1'1'2' or 3'1'1'2'1' or 3'1'2'1'1' or 3'2'1'1'1' or 2'1'1'2'1'1' or 2'1'2'1'1'1' or 2'2'1'1'1'1' or 3'1'1'1'1'1' or 2'1'1'1'1'1'1' or 1'1'1'1'1'1'1'1' where k' = k pi/4. Note 3'2'2'1 and 3'1'2'2'; 3'1'1'2'1' and 3'1'2'1'1'; 3'1'1'1'2' and 3'2'1'1'1' are different by rotation but not reflection
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..200

Crossrefs

Cf. A008965, A091696, A098912.

Programs

PARI

b(n) = (1/n)*sumdiv(n, d, eulerphi(n/d) * 2^d);
a(n) = b(2*n) - 1 - 2^n; \\ Andrew Howroyd, Sep 06 2017

Formula

From Andrew Howroyd, Sep 06 2017: (Start)

a(n) = A008965(2*n) - 2^n.

a(n) = (Sum_{d | 2*n} phi(2*n/d) * 2^d)/(2*n) - 1 - 2^n.

(End)

Extensions

Terms a(8) and beyond from Andrew Howroyd, Sep 06 2017