- Pentatope number
A pentatope number is a number in the fifth cell of any row of
Pascal's trianglestarting 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
pentatopewith side length 5 contains 70 spheres. Each layer represents one of the first five tetrahedral numbers. For example the bottom (green) layer has 35 spheres in total.] Pentatope numbers belong in the class of figurate numbers, which can be represented as regular, discrete geometric patterns. The formula for the "n"th pentatopic number is::.
Two of every three pentatope numbers are also
pentagonal numbers. To be precise, the th pentatope number is always the th pentagonal number and the th pentatope number is always the th pentagonal number. The th pentatope number is the generalized pentagonal number obtained by taking the negative index in the formula for pentagonal numbers. (These expressions always give integers).
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