Suppose that $$c_0+c_1x+c_2x^2+ \cdots = \frac{1} {\lfloor{r}\rfloor+\lfloor{2r}\rfloor x+\lfloor{3r}\rfloor x^2+ \cdots},$$ where $r = (1+ \sqrt{5})/2$. Let $i_0 < i_1 < i_2 < \cdots $ be the numbers $i$ for which $c_i \ge 0.$ Is $i_j = 2j$ for every $j$?

For what other choices of $r$ is $i_j = 2j$ for every $j$?

Suppose that $$c_0+c_1x+c_2x^2+ \cdots = \frac{1} {\lfloor{r}\rfloor+\lfloor{2r}\rfloor x+\lfloor{3r}\rfloor x^2+ \cdots},$$ where $r = (1+ \sqrt{5})/2$. Let $i_0 < i_1 < i_2 < \cdots $ be the numbers $i$ for which $c_i \ge 0.$ Is $i_j = 2j$ for every $j$?

For what other choices of $r$ is $i_j = 2j$ for every $j$?

(Added on Jan. 4, 2017) Here are some initial terms: $$ (c_0,c_1,c_2, \dots) = (1, -3, 5, -9, 17, -30, 52, -90, 154, -262, 446, -758, 1285, \dots ).$$ The first question is this: do the signs alternate for the whole sequence? The second question may lead to some surprises.