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