A remarkable sequence related to $\pi$ and $\sqrt{2}$

TL;DR

This paper proves that five different definitions of a specific integer sequence related to pi and sqrt(2) in OEIS are equivalent, confirming their consistency.

Contribution

It establishes the equivalence of five distinct definitions of the sequence, clarifying its properties and ensuring consistency across different formulations.

Findings

All five definitions produce the same sequence

The sequence relates to pi and sqrt(2)

The proof confirms the sequence's well-defined nature

Abstract

We prove that five ways to define entry A086377 in the On-Line Encyclopedia of Integer Sequences do lead to the same integer sequence.