A098912 Number of ways angles from Pi/n to (n-1)Pi/n can tile around a vertex, where rotations and reflections of an angle sequence are not counted.

1, 5, 16, 54, 180, 607, 2098, 7397, 26452, 95821, 350554, 1292634, 4797694, 17904220, 67125898, 252679320, 954505718, 3616951513, 13744169104, 52358244166, 199912298266, 764879838343, 2932035371786, 11259007784430, 43303859981236, 166800020984581

Offset: 2

Author

Stuart E Anderson, Oct 17 2004

Keywords

• nonn

Comments

Sequence can be interpreted as a tiling of rhombs (with appropriate angles) around a vertex. E.g.. a(5) is the number of ways Penrose rhombs can tile around a vertex.

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

Examples

			a(3) = 5 because we can write 2pi = 2'+2'+2' or 2'+1'+2'+1' or 2'+2'+1'+1' or 2'+1'+1'+1'+1' or 1'+1'+1'+1'+1'+1' where k' = k pi/3.
		

Links

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

Crossrefs

Cf. A005418, A091696, A098913.

Programs

Mathematica

b[n_] := (1/n)*DivisorSum[n, EulerPhi[n/#] * 2^# &];
a[n_] := b[2*n]/2 + 2^(n-2) - 2^Quotient[n, 2] - 2^Quotient[n-1, 2];
Table[a[n], {n, 2, 27}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

PARI

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

Formula

From Andrew Howroyd, Sep 06 2017: (Start)

a(n) = A091696(2*n) - 1 - Sum_{1..n} A005418(n).

a(n) = 2^(n-2) - 2^(floor(n/2)) - 2^(floor((n-1)/2)) + (1/(4*n)) * (Sum_{d | 2*n} phi(2*n/d) * 2^d).

(End)

Extensions

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