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