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

2,2

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

Links

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

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)

Example

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

Prog

(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

Crossrefs

Cf. A008965, A091696, A098912.

Sequence in context: A254686 A295374 A212403 * A126392 A206373 A149767

Adjacent sequences: A098910 A098911 A098912 * A098914 A098915 A098916

Keyword

nonn

Author

Stuart E Anderson, Oct 17 2004

Extensions

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

Status

approved