Let $ f ( x ) \in \mathbb { Z } [ x ] $ be a polynomial with integer coefficients such that $ f ( 1 ) = - 1 , f ( 4 ) = 2 $ and $f ( 8 ) = 34 $. Suppose $n\in\mathbb{Z}$ is an integer such that $ f ( n ) = n ^ { 2 } - 4 n - 18 $. Determine all possible values for $n$.
1. Define the polynomial \( f(x) \) with integer coefficients such that \( f(1) = -1 \), \( f(4) = 2 \), and \( f(8) = 34 \). We are given that \( f(n) = n^2 - 4n - 18 \) for some integer \( n \).

2. Define a new polynomial \( g(x) \) as follows:
   \[
   g(x) = f(x) - (x^2 - 4x - 18)
   \]
   This simplifies to:
   \[
   g(x) = f(x) - x^2 + 4x + 18
   \]
   Since \( f(x) \) has integer coefficients, \( g(x) \) also has integer coefficients.

3. Suppose \( n \) is a root of \( g(x) \), then:
   \[
   g(n) = f(n) - (n^2 - 4n - 18) = 0
   \]
   This implies:
   \[
   f(n) = n^2 - 4n - 18
   \]

4. Since \( g(x) \) has integer coefficients, we can write:
   \[
   g(x) = (x - n_i) h(x)
   \]
   where \( n_i \) is a root of \( g(x) \) and \( h(x) \in \mathbb{Z}[x] \).

5. Evaluate \( g(x) \) at the given points:
   \[
   g(1) = f(1) - (1^2 - 4 \cdot 1 - 18) = -1 - (-21) = 20
   \]
   \[
   g(4) = f(4) - (4^2 - 4 \cdot 4 - 18) = 2 - (-18) = 20
   \]
   \[
   g(8) = f(8) - (8^2 - 4 \cdot 8 - 18) = 34 - 14 = 20
   \]

6. Since \( g(1) = 20 \), \( g(4) = 20 \), and \( g(8) = 20 \), we have:
   \[
   1 - n_i \mid 20
   \]
   \[
   4 - n_i \mid 20
   \]
   \[
   8 - n_i \mid 20
   \]

7. We need to find \( n_i \) such that \( 1 - n_i \), \( 4 - n_i \), and \( 8 - n_i \) all divide 20. Consider the possible values of \( n_i \):
   \[
   1 - n_i = k_1 \implies n_i = 1 - k_1
   \]
   \[
   4 - n_i = k_2 \implies n_i = 4 - k_2
   \]
   \[
   8 - n_i = k_3 \implies n_i = 8 - k_3
   \]

8. The possible values of \( k_1, k_2, k_3 \) that divide 20 are \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \). We need to find common values for \( n_i \) that satisfy all three conditions.

9. Solving for \( n_i \):
   \[
   1 - n_i \mid 20 \implies n_i = 1 - k_1
   \]
   \[
   4 - n_i \mid 20 \implies n_i = 4 - k_2
   \]
   \[
   8 - n_i \mid 20 \implies n_i = 8 - k_3
   \]

10. The common values of \( n_i \) that satisfy all three conditions are:
    \[
    n_i = -5, -2, 2, 1, 5
    \]

11. The corresponding values of \( n \) are:
    \[
    n = 6, 3
    \]

The final answer is \(\boxed{6, 3}\)