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For the record, to answer the question in the OP: The desired PDF $P_n(r)$ of the distance $r$ from the origin for arbitrary $n$ can be expressed as an integral over Bessel functions $J_0$, $$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$ See theorem 2.1 in Densities of short uniform random walks in higher dimensions for a derivation (and the generalization to $d$ dimensions).