Proof by induction for recursive sequence $a_n =a_{\lfloor n/2 \rfloor} + a_{\lceil n/2 \rceil} +3n +1$
Induction with floor, ceiling $n\le2^k\implies a_n\le3\cdot k2^k+4\cdot2^k-1$ for $a_n=a_{\lfloor\frac{n}2\rfloor}+a_{\lceil\frac{n}2\rceil}+3n+1$