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
Examples
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.
Formula
a(p) = (2^(p-1)-1)/p + 2^((p-1)/2) for odd prime p. - Andrew Howroyd, Mar 22 2023
Extensions
a(7) and a(9) corrected and a(11)-a(20) from Andrew Howroyd, Mar 22 2023
Comments