TITLE: Number of distinct normalized vectors from the center of a hexagon in a hexagonal grid
QUESTION [5 upvotes]: 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).

REPLY [2 votes]: Let's  move the centers of the hexagons to the grid $\mathbb{Z}\times\mathbb{Z}$ by an affine transformation, so that the centers of the hexagons at hexagonal distance less than or equal to $n$ are now represented by the more Cartesian-looking (but uglier) set
$$H_n:=\{(x,y)\in\mathbb{Z}\times\mathbb{Z}: |x|\le n, |y|\le n, |x-y|\le n, \}.$$
We can  partition $H_n$ into the area in the axis, a square on the first quadrant and a square on the third quadrant; a triangle in the second and a triangle in the fourth quadrant. Counting separately the corresponding normalized vectors from each set, we get, for your $H(n)$ for $n\ge1$
$H(n)=4+2A(n)+2B(n)$, where
$$A(n):=\{(x,y)\in\mathbb N_+ \times \mathbb N_+ :  x\le n,  y\le n, (x,y)=1\}$$
is easily seen to be $A(n)=  -1+\sum_{k=1}^n \phi(k)=$ A18805 , and so is
$$B(n):=\{(x,y)\in\mathbb N_+ \times \mathbb N_+ :  1\le x<y\le n, (x,y)=1\}$$
which is $B(n)=2\sum_{k=1}^n \phi(k)=$ A15614.
So for $n\le50$ you have
$$6,12,24,36,60,72,108,132,168,192,$$$$252,276,348,384,432,480,576,612,720,
768,$$$$840,900,1032,1080,1200,1272,1380,1452,1620,1668,$$$$1848,1944,2064,
2160,2304,2376,2592,2700,2844,2940,$$$$3180,3252,3504,3624,3768,3900,4176,
4272,4524,4644,4836,4980,5292,$$$$\dots$$
(In particular the first difference is $6\phi(n)$ as was immediately spotted by Steven Stadnicki)