Generalized Pentagonal Numbers and Centered Hexagonal Numbers
Generalized pentagonal numbers are closely related to centered hexagonal numbers. When the array corresponding to a centered hexagonal number is divided between its middle row and an adjacent row, it appears as the sum of two generalized pentagonal numbers, with the larger piece being a pentagonal number proper:
| 1=1+0 | 7=5+2 | 19=12+7 | 37=22+15 | |||
|---|---|---|---|---|---|---|
In general:
where both terms on the right are generalized pentagonal numbers and the first term is a pentagonal number proper (n ≥ 1). This division of centered hexagonal arrays gives generalized pentagonal numbers as trapezoidal arrays, which may be interpreted as Ferrers diagrams for their partition. In this way they can be used to prove the pentagonal number theorem referenced above.
Read more about this topic: Pentagonal Number
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