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The first few numbers of this kind are :
Pentatope numbers belong in the class of figurate numbers, which can be represented as regular, discrete geometric patterns. The formula for the nth pentatopic number is:
Two of every three pentatope numbers are also pentagonal numbers. To be precise, the (3k − 2)th pentatope number is always the ((3k2 − k)/2)th pentagonal number and the (3k − 1)th pentatope number is always the ((3k2 + k)/2)th pentagonal number. The 3kth pentatope number is the generalized pentagonal number obtained by taking the negative index −(3k2 + k)/2 in the formula for pentagonal numbers. (These expressions always give integers).
The infinite sum of the reciprocals of all pentatopal numbers is . This can be derived using telescoping series.
Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.
If the corners of a polygon of size n are all connected to one another, the number of intersections created will be a pentatope number. For example, a triangle has 0 intersections, a square has 1, a pentagon has 5, a hexagon has 15, and a heptagon has 35.
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