My favorite integer sequence

[sheldonison]

.... I think we can generalize to the reals:
f'(x) = f(x/2) for real x

Rewriting as f'(2*x) = f(x), and taking repeated derivatives, you get:

\( \frac{d^n}{{dx}^n} f(2^{n} x) = f(x) \)

This becomes a differential equation. We only need the first derivative to solve numerically (small step size is needed), but a neat recurrence relation leads to the following power series:

\( f(x) = \sum_{k=0}^{\infty}\frac{1}{2^{k(k-1)/2} k!} x^k \)

f(x) in the complex plane, from real -60 to +100 with grids every 20 units
   

f(f(x)) in the complex plane from real -60 to +100, with grids every 20 units. Notice for positive reals, it acts somewhat like exp(x), but the imaginary period is slowly increasing. These two plots can be compared to the equivalent plots in the exp^0.5 post, http://math.eretrandre.org/tetrationforu...863&page=3, where the asymptotic converges much more closely to exp(z), and the asymptotic converges to a 2pi i period, whereas this function has an imaginary cycle that is increasing, but in many other ways, the plots are similar.