## Irrational base

### Problem 558

Let `r` be the real root of the equation `x`^{3} = `x`^{2} + 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` = $\displaystyle \sum_{k=-\infty}^{\infty}$ `b _{k} r^{k}`

under the conditions that:

`b`is 0 or 1 for all

_{k}`k`;

`b`+

_{k}`b`

_{k+1}+

`b`

_{k+2}≤ 1 for all

`k`;

`w`(

`n`) = $\displaystyle \sum_{k=-\infty}^{\infty}$

`b`is finite.

_{k}Let S(`m`) = $\displaystyle \sum_{j=1}^{m}$ `w`(`j`^{2}).

You are given S(10) = 61 and S(1000) = 19403.

Find S(5 000 000).