(In reply to spoiler solution; can it clue toward the intended method?
Add one more sequence adding consecutive pairs of Sigma sums, call it T(n):
n | T(n) = [Sigma(k=1 to n-1) S(k)] + [Sigma(k=1 to n) S(k)]
5 | 17
6 | 31
7 | 57
8 | 105
9 | 193
This is S(n) shifted a few places, specifically T(n) = S(n+2)
Also, any sequence defined with the recursion S(k+3) = S(k+2) + S(k+1) + S(k) will have lim S(k+1)/S(k) = 1.8393..., which is the positive real root of x^3-x^2-x-1 = 0. This is analogous to the Fibonacci sequence limit of 1.618..., the positive real root of x^2-x-1 = 0.
OEIS has S(k) as A000213 and the sums are A001590, which actually has the same recursive generator with different starting values.