An Equidistribution Result for Differences Associated with Square Pyramidal Numbers

We provide an asymptotic formula for the average value of the sequence A351830: $a_n=\mathrm{\mid }{P}_n-y_n^2\mathrm{\mid }$ for $1\le n\le x$, where $P_n$ is the $n$-th square pyramidal number and $y_n^2$ is the closest square to $P_n$. Moreover, we supply asymptotic formulas for the $k$-th moment of the same sequence, for any fixed natural number $k$.