To find the sum \( S = 1 \cdot 4 + 4 \cdot 7 + 7 \cdot 10 + \ldots + 2017 \cdot 2020 \), we need to identify the pattern in the sequence and then derive a formula for the sum.

First, observe the general term in the sequence. The \( n \)-th term can be written as:
\[ a_n = (3n - 2)(3n + 1) \]

Let's verify this pattern with the first few terms:
- For \( n = 1 \): \( a_1 = (3 \cdot 1 - 2)(3 \cdot 1 + 1) = 1 \cdot 4 = 4 \)
- For \( n = 2 \): \( a_2 = (3 \cdot 2 - 2)(3 \cdot 2 + 1) = 4 \cdot 7 = 28 \)
- For \( n = 3 \): \( a_3 = (3 \cdot 3 - 2)(3 \cdot 3 + 1) = 7 \cdot 10 = 70 \)

These match the given terms in the sequence.

Next, we need to find the sum of the sequence up to the term where \( 3n - 2 = 2017 \). Solving for \( n \):
\[ 3n - 2 = 2017 \]
\[ 3n = 2019 \]
\[ n = 673 \]

So, the sequence has 673 terms. The sum \( S \) can be written as:
\[ S = \sum_{n=1}^{673} (3n - 2)(3n + 1) \]

We can expand the general term:
\[ (3n - 2)(3n + 1) = 9n^2 + 3n - 6n - 2 = 9n^2 - 3n - 2 \]

Thus, the sum \( S \) is:
\[ S = \sum_{n=1}^{673} (9n^2 - 3n - 2) \]

We can split this into three separate sums:
\[ S = 9 \sum_{n=1}^{673} n^2 - 3 \sum_{n=1}^{673} n - 2 \sum_{n=1}^{673} 1 \]

Using the formulas for the sums of squares, the sum of the first \( n \) natural numbers, and the sum of constants:
\[ \sum_{n=1}^{k} n^2 = \frac{k(k+1)(2k+1)}{6} \]
\[ \sum_{n=1}^{k} n = \frac{k(k+1)}{2} \]
\[ \sum_{n=1}^{k} 1 = k \]

Substituting \( k = 673 \):
\[ \sum_{n=1}^{673} n^2 = \frac{673 \cdot 674 \cdot 1347}{6} \]
\[ \sum_{n=1}^{673} n = \frac{673 \cdot 674}{2} \]
\[ \sum_{n=1}^{673} 1 = 673 \]

Now, calculate each part:
\[ \sum_{n=1}^{673} n^2 = \frac{673 \cdot 674 \cdot 1347}{6} \]
\[ \sum_{n=1}^{673} n = \frac{673 \cdot 674}{2} = 226051 \]
\[ \sum_{n=1}^{673} 1 = 673 \]

Finally, substitute these into the expression for \( S \):
\[ S = 9 \left( \frac{673 \cdot 674 \cdot 1347}{6} \right) - 3 \cdot 226051 - 2 \cdot 673 \]

Simplify each term:
\[ 9 \left( \frac{673 \cdot 674 \cdot 1347}{6} \right) = 9 \left( \frac{607927962}{6} \right) = 9 \cdot 101321327 = 911891943 \]
\[ 3 \cdot 226051 = 678153 \]
\[ 2 \cdot 673 = 1346 \]

So:
\[ S = 911891943 - 678153 - 1346 \]
\[ S = 911212444 \]

Therefore, the sum \( S \) is:
\[ S = 911212444 \]