A098913 - OEIS

%I #7 Sep 06 2017 20:47:21

%S 1,5,19,75,287,1053,3859,14089,51463,188697,695155,2573235,9571195,

%T 35759799,134154259,505163055,1908619755,7233118641,27486768415,

%U 104713346699,399818311219,1529747101965,5864045590035,22517965253595,86607619323751,333599840675337

%N 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.

%C 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.

%C Also, the number of nonequivalent compositions of 2*n with maximum part size n-1 up to rotation. - _Andrew Howroyd_, Sep 06 2017

%H Andrew Howroyd, <a href="/A098913/b098913.txt">Table of n, a(n) for n = 2..200</a>

%F From _Andrew Howroyd_, Sep 06 2017: (Start)

%F a(n) = A008965(2*n) - 2^n.

%F a(n) = (Sum_{d | 2*n} phi(2*n/d) * 2^d)/(2*n) - 1 - 2^n.

%F (End)

%e 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

%o (PARI)

%o b(n) = (1/n)*sumdiv(n, d, eulerphi(n/d) * 2^d);

%o a(n) = b(2*n) - 1 - 2^n; \\ _Andrew Howroyd_, Sep 06 2017

%Y Cf. A008965, A091696, A098912.

%K nonn

%O 2,2

%A _Stuart E Anderson_, Oct 17 2004

%E Terms a(8) and beyond from _Andrew Howroyd_, Sep 06 2017