Consider an infinite hexagonal grid composed of regular hexagons. Choose any hex to be the origin hex. Let n be a natural number.

Find an expression, in terms of n, for the number of distinct normalized vectors from the center of the origin hex to the center of every hexagon whose distance from the origin (measured in hexagons) is less than or equal to n.

I manually calculated up to n=15:

n | distinct normalized vectors | increment |
---|---|---|

1 | 6 | - |

2 | 12 | 6 |

3 | 24 | 12 |

4 | 36 | 12 |

5 | 60 | 24 |

6 | 72 | 12 |

7 | 108 | 36 |

8 | 132 | 24 |

9 | 168 | 36 |

10 | 192 | 24 |

11 | 252 | 60 |

12 | 276 | 24 |

13 | 348 | 72 |

14 | 384 | 36 |

15 | 432 | 48 |

What I do know:

- The increment from n to n+1 has to be a multiple of 6, because of the simmetry of the hexagon.
- The expression has to be less than 3n^2+3n, since this is the formula for the total number of hexagons at a distance n, at most, from the origin hex (in hexagons);
- When n is prime, the increment from n-1 is 6*(n-1).

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