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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.

%I A098912 #13 Oct 08 2017 10:40:53
%S A098912 1,5,16,54,180,607,2098,7397,26452,95821,350554,1292634,4797694,
%T A098912 17904220,67125898,252679320,954505718,3616951513,13744169104,
%U A098912 52358244166,199912298266,764879838343,2932035371786,11259007784430,43303859981236,166800020984581
%N 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.
%C A098912 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.
%C A098912 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
%H A098912 Andrew Howroyd, <a href="/A098912/b098912.txt">Table of n, a(n) for n = 2..200</a>
%F A098912 From _Andrew Howroyd_, Sep 06 2017: (Start)
%F A098912 a(n) = A091696(2*n) - 1 - Sum_{1..n} A005418(n).
%F A098912 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).
%F A098912 (End)
%e A098912 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.
%t A098912 b[n_] := (1/n)*DivisorSum[n, EulerPhi[n/#] * 2^# &];
%t A098912 a[n_] := b[2*n]/2 + 2^(n-2) - 2^Quotient[n, 2] - 2^Quotient[n-1, 2];
%t A098912 Table[a[n], {n, 2, 27}] (* _Jean-François Alcover_, Oct 08 2017, translated from PARI *)
%o A098912 (PARI)
%o A098912 b(n) = (1/n)*sumdiv(n, d, eulerphi(n/d) * 2^d);
%o A098912 a(n) = b(2*n)/2 + 2^(n-2) - 2^(n\2) - 2^((n-1)\2); \\ _Andrew Howroyd_, Sep 06 2017
%Y A098912 Cf. A005418, A091696, A098913.
%K A098912 nonn
%O A098912 2,2
%A A098912 _Stuart E Anderson_, Oct 17 2004
%E A098912 Terms a(8) and beyond from _Andrew Howroyd_, Sep 06 2017