That's the arithmetic sequence 

7, 20, 33, 46, 59, 72, 85, 98, 111, 124, 137, 150

Make a new arithmetic sequence by subtracting 7 from every term:

0, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143

Make another new arithmetic sequence by dividing every term by 13:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

The smallest possible sum represented as a sum of four distinct 
numbers chosen from this sequence is 0+1+2+3 = 6

The largest possible sum represented by a sum of four distinct 
numbers chosen from it is 8+9+10+11 = 38

Every integer from 6 to 38 inclusive can be represented as a sum of four distinct numbers chosen from the set {0,1,2,3,...,11

That's 38-6+1 = 33 possible sums.  For every one of those sums, there is
a corresponding sum of four distinct numbers chosen from the set {7,20,33,46,...,150}? 

Answer: 33 different integers can be represented as a sum of four distinct numbers chosen from the set {7,20,33,46,...,150}

Edwin