Apparently the solution is

(1) correct an error in the question which renders it unsolvable (a_3 should be 9, not 7)

(2) guess that the invented term "convolution series" doesn't have anything to do with the mathematical concept of convolution[1]

(3) a_n = nth square number minus nth prime minus nth Fibonacci number.

Staring you right in the face!

[1] https://en.wikipedia.org/wiki/Convolution#Definition

Dude. Relax. I said convolution series and not convolution integral ( the one you've linked to ). The "invented term" as you call it appears in Melzak's textbook ( companion to concrete math ). Finally, I actually "invented" a sequence that appears in OEIS ( https://oeis.org/A160138 ) , so I'm well aware that there are infinite sequences you can manufacture by simply changing the op in a convolution series. I just throw these questions into the mix to have some fun and games, otherwise doing interviews becomes very tedious and boring. Nobody is being judged on whether they can figure out that the number of moves for a legitimate tower of hanoi ie. 1,3,7,15, comes from the 2^n-1 sequence ( there's an entire chapter of D.E.Knuth's "concrete math" devoted to just that very topic, and it was my text in undergrad, so I like to have some fun with it. )