You create your staggered arrangement in some double loop and decide randomly
for each position, whether you put a cylinder or not. This directly
translates into code.

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One way doing that staggered arrangement of cylinders...

linear_extrude(40)
for (x = [0 : 9], y = [0 : 9])
if ((x + y) % 2 == 0)
translate(10 * [x, y])
circle(3);