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# Prism numbers

## What are prism numbers?

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

1. a prism volume: a stack of ${\displaystyle \scriptstyle n\,}$ layers of ${\displaystyle \scriptstyle n\,}$th ${\displaystyle \scriptstyle k\,}$-gonal numbers (so that every edge has ${\displaystyle \scriptstyle n\,}$ balls)
2. a prism surface: two end faces of ${\displaystyle \scriptstyle n\,}$th ${\displaystyle \scriptstyle k\,}$-gonal numbers, with the addition of ${\displaystyle \scriptstyle n-2\,}$ balls on the ${\displaystyle \scriptstyle n\,}$ joining edges (so that every edge has ${\displaystyle \scriptstyle n\,}$ balls)
3. a (better) prism surface: two end faces of ${\displaystyle \scriptstyle n\,}$th ${\displaystyle \scriptstyle k\,}$-gonal numbers, where each joining faces are ${\displaystyle \scriptstyle n\,}$th square numbers (with the addition of ${\displaystyle \scriptstyle n-2\,}$ balls on the ${\displaystyle \scriptstyle n\,}$ joining edges, and then adding the ${\displaystyle \scriptstyle (n-2)^{2}\,}$ internal balls (corresponding to the ${\displaystyle \scriptstyle (n-2)\,}$th square numbers) for each of the ${\displaystyle \scriptstyle n\,}$ joining faces)
4. what is the definition that would give "A005915 Hexagonal prism numbers: (n + 1)*(3*n^2 + 3*n + 1)." considering that we have "A000384 Hexagonal numbers: n*(2*n-1)."?

Prism numbers:

• Trigonal prism numbers (triangular prism numbers)
• Tetragonal prism numbers (square prism numbers) (definition 1 would give cube numbers)
• Pentagonal prism numbers
• Hexagonal prism numbers
• Heptagonal prism numbers
• Octagonal prism numbers
• Enneagonal prism numbers
• Decagonal prism numbers
• Hendecagonal prism numbers
• Dodecagonal prism numbers
• ...

## Sequences

A005915 Hexagonal prism numbers: ${\displaystyle \scriptstyle (n+1)(3n^{2}+3n+1),\,n\,\geq \,0.\,}$

{1, 14, 57, 148, 305, 546, 889, 1352, 1953, 2710, 3641, 4764, 6097, 7658, 9465, 11536, 13889, 16542, 19513, 22820, 26481, 30514, 34937, 39768, 45025, 50726, 56889, 63532, 70673, ...}

A000384 Hexagonal numbers: ${\displaystyle \scriptstyle n(2n-1),\,n\,\geq \,0.\,}$

{0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278, ...}

How do we get from the hexagonal numbers to the hexagonal prism numbers?Daniel Forgues 09:23, 17 July 2012 (UTC)