The first few numbers of this kind are :
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).
Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.In biochemistry, they represent the number of possible arrangements of n different polypeptide subunits in a tetrameric (tetrahedral) protein. No prime is the predecessor of a pentatope number, and the largest semiprime which is the predecessor of a pentatope number is 1819.
Test for pentatope numbers
- is triangular number.
then : is perfect square.
- Deza, Elena; Deza, M. (2012), "3.1 Pentatope numbers and their multidimensional analogues", Figurate Numbers, World Scientific, p. 162, ISBN 9789814355483
- Sloane, N. J. A. (ed.). "Sequence 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.
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