An algorithm for generating all possible combinations of sums of elements.

sums_of_elements.java

public List<Integer> sums(int sum, int offset, int[] array) {
    List<Integer> sums = new ArrayList<Integer>(array.length - offset);

    for (int i = offset; i < array.length; ++i) {
        int total = sum + array[i];

        sums.add(total);
        sums.addAll(sums(total, i + 1, array));
    }

    return sums;
}