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Please verify that the following information is correct:

  1. The way to pack 13 spheres (with equal diameter), 12 around a 13th in the center, together so that they are all in contact with each other and with the smallest amount of space left over will require a cubeoctahedron as a container.

  2. If you place more spheres of the same diameter around the outer spheres of this sphere-pack, packing them in and next to each other as closely as possible, it takes a total of 42 to create the next layer of close-packed spheres.

  3. If 1 and 2 are correct, is the sphere-packing rule for cubeoctahedra 10(n to the second power) + 2 where n is the layer-number. The formula tells us how many spheres it takes to fill each layer where n is equal to or greater than 1.

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