The other part of the fibonacci sequence

It is widely known that a closed form for the $n$th fibonacci number exists, namely
$$F_n = \frac{r_1^n - r_2^n}{\sqrt5}\\ \qquad r_1 = \frac{1+\sqrt5}{2}\\ \qquad r_2 = \frac{1-\sqrt5}{2}$$

This closely resembles the way that we use complex conjugates to extract imaginary parts from complex numbers $z$:
$$z = a + bi\\ z^* = a - bi\\ \operatorname{Im}(z) = \frac{z - z^*}{2i}$$

Did you notice how in the fibonacci sequence we divide by the special number $\sqrt5$ and here we are dividing by the special number $i$? This means that, in some sense, $F_n$ is the 'imaginary' part of $r_1^n$.  The 'real' part, then, should be $E_n = r_1^n + r_2^n$ (possibly divided by $2$) by analogy with $\operatorname{Re}(z) = \frac{z + z^*}{2}$.

Is there a recursive interpretation of $E_n$? Apparently, it's the same as for the fibonacci sequence:
n  | 0  1  2  3  4  5  6  7  8  9
E_n| 2  1  3  4  7  11 18 29 47 76