A356444 -id:A356444 - cp's OEIS frontend

A356663 Number of ways to create an angle excess of n degrees using 3 distinct regular polygons with integral internal angles.

0, 1, 3, 1, 3, 5, 1, 3, 4, 5, 2, 7, 2, 5, 6, 4, 2, 6, 2, 4, 5, 4, 2, 5, 4, 4, 6, 5, 2, 7, 2, 5, 6, 4, 6, 7, 4, 6, 9, 7, 5, 9, 6, 9, 9, 8, 6, 10, 6, 7, 8, 6, 6, 8, 6, 5, 7, 6, 4, 10, 3, 7, 7, 7, 7, 10, 6, 6, 10, 9, 7, 9, 6, 9, 11, 10, 7, 10

Offset: 1

Joseph C. Y. Wong, Aug 21 2022

nonn

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.

			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).

Cf. A356444 (where polygons do not have to be distinct).

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)

cp's OEIS Frontend

A356663 Number of ways to create an angle excess of n degrees using 3 distinct regular polygons with integral internal angles.

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