Pentatope number

A pentatope number is a number in the fifth cell of any row of Pascal'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 :

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 $4 \over 3$ $4 \over 3$ . This can be derived using telescoping series.

Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.

then : $8\times {\frac {{\sqrt {24n+1}}+1}{2}}+1$ $8\times {\frac {{\sqrt {24n+1}}+1}{2}}+1$ is perfect square.

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