Nesting Patterns Algorithms / Cutting Waste Problem

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 开发者_Go百科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!

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?

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