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) = 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-François 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
KEYWORD
nonn
AUTHOR
Stuart E Anderson, Oct 17 2004
EXTENSIONS
Terms a(8) and beyond from Andrew Howroyd, Sep 06 2017
STATUS
approved