\[ \text{Let } g(n) = \left\lbrace \begin{array}{lr} \frac23 n & : n \equiv 0 \pmod{3} \\ \frac13 (4n-1) & : n \equiv 1 \pmod{3} \\ \frac13 (4n+1) & : n \equiv 2 \pmod{3} \end{array} \right. \]

Define a recursive sequence \(\{x_n\}\) as \[ \begin{align} x_0 &= 8 \\ x_n &= g(x_{n-1}) \text{ for } n \ge 1 \end{align} \]

This sequence appears to grow exponentially, but no proof is known.

Here is a graph of the base 10 logarithms of the first 1000 terms of the sequence. The equation of the regression line is \(y=1.97034+0.02342x\).