A361635 - OEIS

%I #19 Apr 28 2023 16:23:19

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

%T 30820

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

%F a(p) = (2^(p-1)-1)/p + 2^((p-1)/2) for odd prime p. - _Andrew Howroyd_, Mar 22 2023

%e 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.

%Y Cf. A164896, A361659.

%K nonn,more

%O 1,3

%A _Roman Mecholsky_, Mar 18 2023

%E a(7) and a(9) corrected and a(11)-a(20) from _Andrew Howroyd_, Mar 22 2023