An Euler-Fibonacci sequence (Q2880087)

In the paper under review, the authors introduce the Euler--Fibonacci sequence as the sequence \((X_n)_{n\geq 0}\) given by \(X_0=0,~X_1=1\) and \(X_{n+2}=\varphi(X_{n+1}+X_n+1)\) for all \(n\geq 0\). Here, \(\varphi(m)\) is the Euler function of the positive integer \(m\). They show that \(X_n\) is even for all \(n\geq 3\) and that \(X_n\leq F_n\), where \(F_n\) is the \(n\)th Fibonacci number. The paper concludes with a number of interesting questions like are there infinitely many \(n\) such that \(X_{n+1}<X_n\), and what can one say about the behavior of the quotient \(X_{n+1}/X_n\) as \(n\) tends to infinity.