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Centered prism numbers

What are centered prism numbers?

Right and uniform $k\,$ -gonal centered prism numbers might be defined as ... (has yet to be determined...)

1. a prism volume: a stack of $n\,$ layers of $n\,$ th centered $k\,$ -gonal numbers (so that every edge has $n\,$ balls)
2. a prism surface: two end faces of $n\,$ th centered $k\,$ -gonal numbers, with the addition of $n-2\,$ balls on the $n\,$ joining edges (so that every edge has $n\,$ balls)
3. a (better) prism surface: two end faces of $n\,$ th centered $k\,$ -gonal numbers, where each joining faces are $n\,$ th centered square numbers (with the addition of $n-2\,$ balls on the $n\,$ joining edges, and then adding the $(n-1)^{2}+(n)^{2}\,$ internal balls (corresponding to the $(n-1)\,$ th centered square numbers) for each of the $n\,$ joining faces)
4.  ?

Centered prism numbers:

• Centered trigonal prism numbers (centered triangular prism numbers)
• Centered tetragonal prism numbers (centered square prism numbers)
• Centered pentagonal prism numbers
• Centered hexagonal prism numbers
• Centered heptagonal prism numbers
• Centered octagonal prism numbers
• Centered enneagonal prism numbers
• Centered decagonal prism numbers
• Centered hendecagonal prism numbers
• Centered dodecagonal prism numbers
• ...