ILL 1971 Problem 34

Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and $$T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k (k \geq 3).$$

Show that for all $k$, $$1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \text{and} 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],$$

where $[x]$ denotes the greatest integer not exceeding $x.$