Consider the configurations of integers
$a_{1,1}$
$a_{2,1} \quad a_{2,2}$
$a_{3,1} \quad a_{3,2} \quad a_{3,3}$
$\dots \quad \dots \quad \dots$
$a_{2017,1} \quad a_{2017,2} \quad a_{2017,3} \quad \dots \quad a_{2017,2017}$
Where $a_{i,j} = a_{i+1,j} + a_{i+1,j+1}$ for all $i,j$ such that $1 \leq j \leq i \leq 2016$.
Determine the maximum amount of odd integers that such configuration can contain.
1. **Restate the Problem with General \( n \)**:
   We are given a triangular array of integers \( a_{i,j} \) where \( 1 \leq j \leq i \leq n \). The integers satisfy the relation:
   \[
   a_{i,j} = a_{i+1,j} + a_{i+1,j+1}
   \]
   for all \( 1 \leq j \leq i < n \). We need to determine the maximum number of odd integers in this configuration.

2. **Claim**:
   The maximum number of odd integers in the configuration is given by:
   \[
   \left \lceil \frac{n(n+1)}{3} \right \rceil
   \]

3. **Construction**:
   Define \( a_{i,j} \) such that:
   \[
   a_{i,j} = (3 \nmid (i+j)) \mod 2
   \]
   This means \( a_{i,j} \) is odd if \( i+j \) is not divisible by 3, and even otherwise. This construction satisfies the given additive constraint modulo 2.

4. **Upper Bound**:
   The key idea is that it is impossible for \( a_{i,j} \), \( a_{i+1,j} \), and \( a_{i+1,j+1} \) to all be odd for \( 1 \leq j \leq i < n \). Therefore, every such triangle must contain at least one even number.

5. **Counting Evens**:
   We need to show that there are at least:
   \[
   \frac{n(n+1)}{2} - \left \lceil \frac{n(n+1)}{3} \right \rceil = \left \lfloor \frac{n(n+1)}{6} \right \rfloor
   \]
   even numbers in the array. This can be checked for small values of \( n \) (e.g., \( n < 3 \)).

6. **Inductive Step**:
   For \( n \geq 3 \), we proceed by induction. Assume the statement holds for \( n-3 \). We need to show that the remaining band of three rows (i.e., rows \( n-2, n-1, n \)) requires:
   \[
   \left \lfloor \frac{n(n+1)}{6} \right \rfloor - \left \lfloor \frac{(n-3)(n-2)}{6} \right \rfloor = n-1
   \]
   even numbers.

7. **Base Case**:
   Check the base case for \( n \leq 4 \). For these small values, manually verify that the number of even numbers matches the required count.

8. **Inductive Hypothesis**:
   Assume the hypothesis holds for \( n-3 \). Consider the band of three rows \( n-2, n-1, n \). If \( a_{n-1,2} \) is even, then one of \( a_{n,1} \), \( a_{n-1,1} \), or \( a_{n,2} \) must also be even, giving us two evens. We can then apply the inductive hypothesis to the remaining part of the array.

9. **Conclusion**:
   By induction, the array requires:
   \[
   \left \lfloor \frac{n(n+1)}{6} \right \rfloor
   \]
   even numbers, confirming that the maximum number of odd numbers is:
   \[
   \left \lceil \frac{n(n+1)}{3} \right \rceil
   \]

The final answer is \( \boxed{ \left \lceil \frac{n(n+1)}{3} \right \rceil } \)