Asymptotic growth of a series
$$
\begin{align*}
\sum_{k=1}^{c \log n - 1} k 2^{- \frac{k}{3}} &amp;\le
\sum_{k=1}^{c \log n } k 2^{- \lfloor \frac{k}{3} \rfloor } \le
\sum_{k=1}^{\lceil \frac{c}{3} \log n \rceil} 3k 2^{-k+1} \le
6\sum_{k=0}^{\infty} k 2^{-k} \\ &amp;\le 
6\sum_{k=0}^{\infty} \left( \frac{3}{2} \right)^k 2^{-k} =
6 \sum_{k=0}^{\infty} \left( \frac{3}{4} \right)^k = 6 \cdot 4 = 24 \in O(1).
\end{align*}
$$