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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

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

LINKS

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

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)

EXAMPLE

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.

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-Fran├žois Alcover, Oct 08 2017, translated from PARI *)

PROG

(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

CROSSREFS

Cf. A005418, A091696, A098913.

Sequence in context: A108300 A041469 A089102 * A299685 A268225 A120343

Adjacent sequences:  A098909 A098910 A098911 * A098913 A098914 A098915

KEYWORD

nonn

AUTHOR

Stuart E Anderson, Oct 17 2004

EXTENSIONS

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

STATUS

approved

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Last modified November 25 09:29 EST 2020. Contains 338623 sequences. (Running on oeis4.)