If you look at one of the comments on A053445 you'll see it mentions that the second difference of partition numbers is equivalent to the number of integer partitions with two largest parts equal and each part $\geq 2$. That is, if q(n) is the number of partitions with those properties, then:
$( p(n)-p(n-1) )~-~( p(n-1)-p(n-2) ) = p(n)-2p(n-1)+p(n-2) = q(n)$
for $n \geq 3$. I looked at the references on the page, as well as a couple journal searches, and didn't see anything where this result was originally found. I've discovered this and something similar and would like to refer to him or her for this result.