Symbolic accumulation

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You could define

def sums(a,nmax):
    return [sum(a[i] for i in xrange(n+1)) for n in xrange(nmax)]

Then you could compute for instance:

sage: a = [i^2 for i in xrange(20)]
sage: sums(a,10)
[0, 1, 5, 14, 30, 55, 91, 140, 204, 285]

Is that what you want, or do you want something that would take "symbolic lists" given by their names, and return a symbolic expression consisting in the formal sums of list elements? I'm not sure how one could do that.

Do you just want syntax to add the first k entries in a list? As @kcrisman alluded, you can get the first k entries by slicing, and then add them with

add(a[:k+1])

So you could use it like this:

sage: a = [n for n in range(5)]   # a is a list
sage: a
[0, 1, 2, 3, 4]
sage: seq = [add(a[:k+1]) for k in range(len(a))]
sage: seq
[0, 1, 3, 6, 10]

This works as long as the entries of your list can be added. Here's an example with symbolic variables:

sage: a = [var('a'+str(n)) for n in range(5)]
sage: a
[a0, a1, a2, a3, a4]
sage: seq = [add(a[:k+1]) for k in range(len(a))]
sage: seq
[a0, a0 + a1, a0 + a1 + a2, a0 + a1 + a2 + a3, a0 + a1 + a2 + a3 + a4]