Consecutive Triplets not Sum of Pentagonal Numbers

Theorem
The following triplets of consecutive positive integers are such that none is the sum of $3$ pentagonal numbers:
 * $\left({19, 20, 21}\right)$
 * $\left({88, 89, 90}\right)$
 * $\left({99, 100, 101}\right)$
 * $\left({111, 112, 113}\right)$