How to count densely packed balls from the center in a spiral?

Question

Mathematics

I need to mark the canvas from the center to the outskirts, placing marks with equal density.
shown. that the ideal arrangement is a honeycomb, a billiard pyramid .
M tag requirements

• Ball[M], where each ball is labeled with exactly one label and each label corresponds to exactly one ball.
• Criterion of distance from the center. If M>K, then (Ball[M + 1] - Ball[1]) >= (Ball[K] - Ball[1])
• Construction method. Recursive transformation Shar[M + 1] = Inductor(Shar[M]) of complexity O(1), ideally linear. If this is not resolved, then Shar[M] = Constructor(M)
  

I have seen algorithms for the Archimedes spiral and the logarithmic one. In the logarithmic, it is easier to build a recursive step because, passing an equal length of the spiral, we always move away from the center by an equal distance. Both spirals do not guarantee uniform packing. It is possible to build a broken spiral according to the cells of the coordinate grid, but the packing density will be less than the honeycomb structure. Can the algorithm for cell packing be solved? PS On the recommendation of the resource, I am moving the
clarifications from the comments to the body of the question Something
like this and I need a particular solution to the problem
https://en.wikipedia.org/wiki/Problem_of_packing_in...