In python, I have a list w of integers, and I would like to efficiently generate all lists of integers such that their sum with weights in w is equal to a certain value a, that is

sum([i*j for i,j in zip(w,f)]) == a

ie the product of w with f is fixed by a. Here everything is int valued. The brute force version would be to use itertools:

from itertools import product

a = 20
w = [1,1,3,3]
rslt = []
for f in product(range(a), repeat=len(w)):
   if sum([i*j for i,j in zip(w,f)]) == a:
      rslt.append(list(f))

print(rslt)

This however scales very badly with the value of a. Is there a way of making generating rslt efficiently? I couldn’t find a way to impose constraint with itertools directly. Mathematically, starting with a known vector w, I want to find all vectors f such that their dot product is fixed by a.

I came up with the idea of a recursive function that compute the difference from a at a given stage of the loop, but I’m having trouble with the implementation. If it’s possible to have an implementation that also does arbitrary constraints (e.g. quadratic), that would be fantastic.