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:
 * $\tuple {19, 20, 21}$
 * $\tuple {88, 89, 90}$
 * $\tuple {99, 100, 101}$
 * $\tuple {111, 112, 113}$