OFFSET
1,3
COMMENTS
a(n) is the number of positive integer triples (a, b, c) (not including permutations) and with a, b, c distinct that satisfy n+360 = (a-2)*180/a + (b-2)*180/b + (c-2)*180/c.
For n >= 175, a(n) = 0. This can be proved. The maximum sum of 3 integral internal angle is of a 360-gon, a 180-gon and a 120-gon with internal angles 179, 178 and 177 degrees respectively. Therefore 179+178+177-360 = 174 degrees is the maximum possible angle excess.
EXAMPLE
For n = 1, there are no possible ways to create an angle excess of 1 degree therefore a(1) = 0.
For n = 3, there are 3 possible ways to create an angle excess of 3 degrees. (3-gon, 8-gon, 30-gon), (4-gon, 5-gon, 24-gon), (5-gon, 6-gon, 8-gon).
PROG
(Python)
import itertools
def subs(l):
res = []
for combo in itertools.combinations(l, 3):
res.append(list(combo))
return res
l = [3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360] # Number of sides of polygons with an integral internal angle
for n in range(1, 200):
k = 0
for i in subs(l):
if n + 360 == (i[0] - 2)*180/i[0] + (i[1] - 2)*180/i[1] + (i[2] - 2)*180/i[2]:
k += 1
print(k)
CROSSREFS
KEYWORD
nonn
AUTHOR
Joseph C. Y. Wong, Aug 21 2022
STATUS
approved