- Pentatope number
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**pentatope number**is a number in the fifth cell of any row ofPascal's triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left. The first few numbers of this kind are :

1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365 OEIS|id=A000332[

pentatope with side length 5 contains 70 spheres. Each layer represents one of the first fivetetrahedral number s. For example the bottom (green) layer has 35 spheres in total.] Pentatope numbers belong in the class offigurate number s, which can be represented as regular, discrete geometric patterns. The formula for the "n"th pentatopic number is::$$n + 3} choose 4} = frac{n(n+1)(n+2)(n+3)}{24}.Two of every three pentatope numbers are also

pentagonal number s. To be precise, the $(3k-2)$th pentatope number is always the $((3k^2-k)/2)$th pentagonal number and the $(3k-1)$th pentatope number is always the $((3k^2+k)/2)$th pentagonal number. The $3k$th pentatope number is the generalized pentagonal number obtained by taking the negative index $-(3k^2+k)/2$ in the formula for pentagonal numbers. (These expressions always give integers).

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