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 :
![](../I/m/Pentatope_of_70_spheres_animation.gif)
Pentatope numbers belong in the class of figurate numbers, which can be represented as regular, discrete geometric patterns.[1] 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).[2]
The infinite sum of the reciprocals of all pentatopal numbers is .[3] This can be derived using telescoping series.
Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.[2]
Test for pentatope numbers
- is triangular number.
then : is perfect square.
References
- ↑ Deza, Elena; Deza, M. (2012), "3.1 Pentatope numbers and their multidimensional analogues", Figurate Numbers, World Scientific, p. 162, ISBN 9789814355483
- 1 2 "Sloane's A000332". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Rockett, Andrew M. (1981), "Sums of the inverses of binomial coefficients" (PDF), Fibonacci Quarterly, 19 (5): 433–437. Theorem 2, p. 435.