An improvement to a theorem of Leonetti and Luca (Q6564333)

In the paper under review, the authors prove the following theorem about integer sequences generated by Euler's totient function, which is the main result in the paper. \N\NTheorem 1. Fix an even integer \( k\ge 0 \). The integer sequence \( (x_n)_{n\ge 1} \) defined by \( x_{n+2}=\phi(x_{n+1})+\phi(x_n)+k \), where \( x_1,x_2\ge 1 \) is bounded by \( 2^{2X^2+X-3} \), where \( X=x_1+x_2+2k \).\N\NTheorem 1 gives a nice improvement of the result of \textit{P. Leonetti} and \textit{F. Luca} in [Bull. Aust. Math. Soc. 109, No. 2, 206--214 (2024; Zbl 1540.11009)].