Could you add details on how to make a single loop in this case to calculate values without recursion?

Ok, I hope that the OEIS editors will treat your opinion as authoritative, in case I want to add my code to the sequences under consideration.

Thank you for answer! Could you please add the details of the equivalence?

@PeterTaylor, thank you for comment! Done. Here $L$ is the number of $0$s in the binary expansion of $2n$.

@DaveBenson, thank you for comment! Unfortunately, I am blocked in the encyclopedia. :)

@MaxAlekseyev, please note that I am not a mathematician, but just an experimenter. Most of my experiments are random. OEIS is my best friend in the sense that I just slightly change something in some natural result and often find a simpler formula or algorithm for an already known sequence. Of course, I want to share such findings, marking them as true. To calm my conscience, I ask questions here and additionally add links to proofs, if anyone provide any.

@MaxAlekseyev, thank you for comment! I absolutely agree with you, but at the same time, numerical verification is enough for me (usually up to $10^6$ plus a few random terms for significantly large indexes).