Nesting Patterns Algorithms / Cutting Waste Problem

Question

First off, I'd like to say that I'm already looking into the "Cutting Stock Problem" algorithm, however I feel that I need a bit more clarification, and possibly some help with some of the math (Not my strong point).

What I need to do is have an offset pattern that causes the circle to fit in the crevasse created by having the two circular dies sitting next to each other on the previous line, like so:

O O O O O O
 O O O O O
O O O O O O

I'm hoping someone can point me towards the right algorithm for this! Thanks!

Answer 1

Assuming unit circles, the centers will be at points (2*i + j, sqrt(3)*j) for i and j integers.

Answer 2

If you have three circles of radius r packed together so that they are touching each other, their centers build an equilateral triangle with sides of length 2*r:

 O
O O

The height of the triangle is sqrt(3)*r, so the the circle in the upper row is moved r to the left and sqrt(3)*r to the top, compared to the left circle in the bottom row.

distance r
   ||
   |O         O---- distance
   O O       O O--- sqrt(3)*r

Is this what you are asking for?