a(n) - e^2(n+1)^2 is bounded. Conjecture: there's a constant c = 2.462... such that a(n) = floor(e^2*(n+1)^2-c).

I believe (n+2)^(n+2)/n^n = e^2(n^2 + 2n + 2/3 + o(1)), so the conjecture should be correct (with perhaps finitely many exceptions) with c = e^2/3 = 2.463.... - Charles R Greathouse IV, May 07 2013