I found this method to be much faster than some other methods...

Does it also give the same results?

It is a well-known problem and many approaches are available on the web, look for Kadane's algorithm.

Below is a functional style implementation, see this link for the derivation

In[1]:= data = {1, 2, -4, 1, 3, -2, 3, -1};
        f[{y_, z_}, x_] := {Max[{0, x + y}], Max[Max[0, x + y], z]}
        Last[Fold[f, {0, 0}, data]]

Out[3]= 5

Inspired by Gustavo's use of Partition (it works like a champ) ... here is the over-enumerated, but "functional paradigm", solution.

data={1,2,-4,1,3,-2,3,-1}; Max[Total /@ Flatten[Table[ Partition[data, i, j], {i, Length[data]}, {j, Length[data]}], 2]]

So, on this (puzzle) example, this solution (gets the right answer, 5) but it generates 121 subsequences in doing so. Using the procedural approach one need examine only a handful of subsequences though with lots of conditional statement executions. (Though I haven't, yet, formulated the worst-case for that approach.)

Still asking, is this - functional shortfall - intrinsic?

Thank you.

First, in response to David's question, the key aspect of the problem definition is the subsequence notion - namely that a subsequence is a consecutive list of 1 or more elements, and not just any subset (as Gustavo observes).

Second, in response to Gustavo's interesting suggestion, it's missing some cases - the maximum sum might come from a single (large positive) number, for example. Further, the point of this particular puzzle is to avoid over-enumerating.

That said, I'm completely in favor of the over-enumeration one often gets with the use of the functional programming paradigm. I'm just asking if there's a functional approach to this puzzle that doesn't over-enumerate. Or, is this puzzle inherently sequential, and thus centrally "procedural"?

What are the rules for this problem? The way I read it, the answer is simply the sum of the positive entries (which would give the answer as 1+2+1+3+3). I am guessing that this is not the interpretation that you are using in getting the answer 1+3-2+3.