Would anybody explain how to get to this
If someone ever comes up again with this question here is the answer:
Lets denote F(n) the sequence of numbers to be constructed that sum up to n.
It can BEGIN either with 1, 2, 3 or 4
When it begins with 1, then the answer is 1..F(n-1).. where F(n-1) is also a sequence of numbers that can begin with either 1, 2, 3 or 4 (that is why we include all possible arrangements). The same thing happens when the sequence begins with 2, 3 or 4.
For example if we take N = 5. Then we have: 1..F(4).. or 2..F(3).. or 3..F(2) or 4..F(4).. We can be sure that, for example, F(4) includes all possible sequences that sum up to 4 because we find it from the smaller problems with n < 4 and we know also that this sequence can begin with either 1, 2, 3 or 4 at the time of finding F(5).
Thus we include all possible arrangements of the sequence which elements sum up to n. Written generally and denoting f(n) as a number of such suquences that sum up to n we get such relation: f(n) = 2*f(n-1) + f(n-2) + f(n-3) as it was described earlier in this thread.