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A098913
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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.
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2
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1, 5, 19, 75, 287, 1053, 3859, 14089, 51463, 188697, 695155, 2573235, 9571195, 35759799, 134154259, 505163055, 1908619755, 7233118641, 27486768415, 104713346699, 399818311219, 1529747101965, 5864045590035, 22517965253595, 86607619323751, 333599840675337
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OFFSET
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2,2
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COMMENTS
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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
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 2..200
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FORMULA
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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)
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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Cf. A008965, A091696, A098912.
Sequence in context: A254686 A295374 A212403 * A126392 A206373 A149767
Adjacent sequences: A098910 A098911 A098912 * A098914 A098915 A098916
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KEYWORD
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nonn
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AUTHOR
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Stuart E Anderson, Oct 17 2004
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EXTENSIONS
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Terms a(8) and beyond from Andrew Howroyd, Sep 06 2017
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STATUS
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approved
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