Fit 2600 equally spaced points on concentric circles My friend is working on an art project where she wants to draw 2600 dots on a circular table, symbolising the 2600 deaths of the conflict in east Ukraine. She approached me to solve this, but I've run out of ideas.
The points must be placed in rings, where the space between the rings is equal to the space between the dots in the rings. At the centre of the table should be the zeroth ring which is a single dot. The table is 1.1 metres in diameter, however use a diameter value of 1 metre to leave a margin around the edge of the table. How many rings should there be, how many dots should there be in each ring and what is the value of the spacing for both the space between the dots and the space between the rings? (There may be multiple solutions.)
An exact solution is preferable but approximate solutions are also accepted.
 A: Let $N$ be the number of rings. The distance between rings (and between consecutive points on a ring) is $1/N$. The circumference of the $k$-th ring is $2\,k\,\pi/N$ and holds $2\,k\,\pi$ points (I know it is not an integer, but let's forget it for a moment). The total number of points is then $\sum_{k=1}^N2\,k\,\pi=N(N+1)\,\pi$. Solving $N(N+1)\,\pi=2599$ we get $N=28.26$. So the number of rings should be $28$ or $29$. This is in close agreement with Abstractions's answer. The number of points in the $k$-th ring should be $\lfloor2\,k\,\pi\rfloor$ or $\lceil2\,k\,\pi\rceil$. A possible distribution with $28$ rings is 
$$
6, 12, 18, 25, 31, 38, 44, 51, 57, 63, 70, 76, 82, 88,\\ 95, 101, 107, 114, 120, 126, 132, 139, 145, 151, 158, 164, 170, 176
$$
This adds up to $2559$ points, so we still have to add $40$ points, in such a way that the distance between points changes as little as possible. This implies adding more points in the larger rings. Distributing them proportionally to the original distribution $\lfloor2\,k\,\pi\rfloor$ I get
$$
6, 12, 18, 25, 31, 38, 44, 51, 57, 63, 71, 77, 83, 89,\\ 96, 103, 109,
116, 122, 128, 134, 142, 148, 154, 161, 167, 174, 180
$$
A: 
The points must be placed in rings, where the space between the rings is equal to the space between the dots in the rings. At the centre of the table should be the zeroth ring which is a single dot.

$0^\text{th}$ ring - 1 point. $1^\text{st}$ ring (radius 1) then must contain exactly 6 points. $2^\text{nd}$ ring - 12 points since radius is 2 and distance is the same. All in all, $n$ rings give you $3*n*(n-1)+1$ points, closest to $2600$ is $n=30$, $2611$ points.
