Irrational base

Published on Saturday, 30th April 2016, 01:00 pm; Solved by 184;
Difficulty rating: 65%

Problem 558

Let r be the real root of the equation x3 = x2 + 1.
Every positive integer can be written as the sum of distinct increasing powers of r.
If we require the number of terms to be finite and the difference between any two exponents to be three or more, then the representation is unique.
For example, 3 = r -10 + r -5 + r -1 + r 2 and 10 = r -10 + r -7 + r 6.
Interestingly, the relation holds for the complex roots of the equation.

Let w(n) be the number of terms in this unique representation of n. Thus w(3) = 4 and w(10) = 3.

More formally, for all positive integers n, we have:
n = $\displaystyle \sum_{k=-\infty}^{\infty}$ bk rk
under the conditions that:
bk is 0 or 1 for all k;
bk + bk+1 + bk+2 ≤ 1 for all k;
w(n) = $\displaystyle \sum_{k=-\infty}^{\infty}$ bk is finite.

Let S(m) = $\displaystyle \sum_{j=1}^{m}$ w(j2).
You are given S(10) = 61 and S(1000) = 19403.

Find S(5 000 000).