Proving complexity of $T(n)=2T(n/3 + 1) + n$ non-Akra-Bazzi
You can use the variant by Leighton "Notes on Better Master Theorems for Divide-and-Conquer Recurrences". Lehman, Leighton and Meyer in "Mathematics for Computer Science"
on page 1019 state a simplified version: If you have $T(n) = \sum_{1 \le i \le k} a_i T(b_i n + h_i(n)) + g(n)$, where $a_i &gt; 0$, $0 &lt; b_i &lt; 1$, $g(n) &gt; 0$ such that $\lvert g'(x) \rvert$ is bounded by a polynomial, and $\lvert h_i(n) \rvert = O(n / \log^2 n)$, then is $p$ is the unique real such that $\sum_{1 \le i \le k} a_i b_i^p = 1$, then:

$$
T(n) = \Theta\left(
               n^p \left(
                      1 + \int_1^n \frac{g(u)}{u^{p + 1}} d u
                   \right)
             \right)
$$

Your $g(n) = n$ certainly qualifies. With $a_1 = 2$, $b_1 = 1/3$, $h_1(n) = 1$ satisfies the condition, you get:

$$
  2 \cdot 3^{-p} = 1
$$

from which $p = \log_3 2$, so that:

$$
T(n) = \Theta\left(
               n^{\log_3 2}
                  \left(
                      1 + \int_1^n u^{- \log_3 2} d u
                   \right)
             \right)
     = \Theta(n)
$$