A361635 Number of strictly-convex unit-sided polygons with all internal angles equal to a multiple of Pi/n, ignoring rotational and reflectional copies.

0, 1, 3, 4, 7, 16, 17, 28, 70, 85, 125, 392, 379, 704, 3359, 2248, 4111, 18510, 14309, 30820

Offset

1,3

Links

Table of n, a(n) for n=1..20.

Formula

a(p) = (2^(p-1)-1)/p + 2^((p-1)/2) for odd prime p. - Andrew Howroyd, Mar 22 2023

Example

For n=3, a(3) is computed as follows:  The base angle is Pi/3 (60 degrees).  Thus any internal angle can only be either Pi/3 or 2*Pi/3.  Call an interior angle with Pi/3 a "1" and with 2*Pi/3 a "2".  Since all external angles will add to 2*Pi, we know that the only possible sequences (ignoring rotation and reflection) are {{1, 1, 1}, {1, 1, 2, 2}, {1, 2, 1, 2}, {1, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2}}.  However, neither {1, 1, 2, 2} nor {1, 2, 2, 2, 2} forms a closed polygon.  Thus the final set is {{1, 1, 1}, {1, 2, 1, 2}, {2, 2, 2, 2, 2, 2}}, which gives a(3) = 3.

Crossrefs

Cf. A164896, A361659.

Sequence in context: A187493 A027020 A130755 * A286348 A116090 A287741

Adjacent sequences: A361632 A361633 A361634 * A361636 A361637 A361638

Keyword

nonn,more

Author

Roman Mecholsky, Mar 18 2023

Extensions

a(7) and a(9) corrected and a(11)-a(20) from Andrew Howroyd, Mar 22 2023

Status

approved