A356663 -id:A356663 - cp's OEIS frontend

A356444 Number of ways to create an angle excess of n degrees using 3 regular polygons with integral internal angles.

0, 1, 3, 1, 3, 6, 1, 3, 4, 6, 2, 9, 2, 5, 7, 5, 2, 9, 2, 6, 6, 4, 2, 8, 4, 5, 7, 7, 2, 12, 3, 6, 7, 5, 7, 10, 4, 6, 9, 10, 5, 12, 6, 10, 11, 8, 6, 14, 6, 11, 9, 8, 6, 12, 8, 7, 8, 8, 5, 15, 3, 7, 8, 8, 7, 12, 6, 8, 10, 12, 7, 14, 6, 10, 13

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) that satisfy n+360 = (a-2)*180/a + (b-2)*180/b + (c-2)*180/c.

For n >= 178, a(n) = 0. This can be proved. The maximum integral internal angle is of a 360-gon with internal angle 179 degrees. Therefore 179*3-360 = 177 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. A356663 (where distinct polygons are allowed).

import itertools
def subs(l):
    res = []
    for combo in itertools.combinations_with_replacement(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

A356444 Number of ways to create an angle excess of n degrees using 3 regular polygons with integral internal angles.

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