Equivalent conditions for the $n$th element of the Beatty sequence $B_{\sqrt{2}}$ being even

TL;DR

This paper establishes conditions under which the nth element of the Beatty sequence for √2 is even, linking two OEIS sequences and enhancing understanding of their properties.

Contribution

It provides new equivalent conditions for the parity of sequence elements and shows the equivalence of two OEIS sequences related to Beatty sequences.

Findings

Identified conditions for even elements in the Beatty sequence B_{√2}

Proved the equivalence of OEIS sequences A090892 and A120752

Enhanced understanding of the structure of Beatty sequences

Abstract

We provide equivalent conditions for the $n$th element of the Beatty sequence $B_{\sqrt{2}}$ being even. In particular, we show that the integer sequences A090892 and A120752 in the OEIS are essentially identical.