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December 2010

PrefixSet edition (2010-10)

Find the shortest addition chain ending with a given integer.

Do you like puzzles? Here is one for you. You are given a positive integer A. The goal is to construct the shortest possible sequence of integers ending with A, using the following rules:

- the first element of the sequence is 1,
- each of the successive elements is the sum of any two preceding elements (adding a single element to itself is also permissible),
- each element is larger than all the preceding elements; that is, the sequence is increasing.

For example, for A = 42, a possible solution is `[1, 2, 3, 6, 12, 24, 30, 42]`. Another possible solution is `[1, 2, 4, 5, 8, 16, 21, 42]`.

Write a function:

class Solution { public int[] solution(int A); }

that, given an integer A, returns the shortest possible sequence of integers satisfying the above conditions and ending with A.

Result array should be returned as an array of integers.

For example, given A = 42, the function may return the sequence `[1, 2, 3, 6, 12, 24, 30, 42]`, as explained above.

Write an ** efficient** algorithm for the following assumptions:

- A is an integer within the range [1..600].

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