Pensieve header: The $p$-tower Hanoi problem, following Grosu’s arXiv://1508.04272.

Φ[3,n_] := Φ[3,n] = (2^n)-1;
Φ[p_,1] = 1;
Φ[p_,n_] /; p>3∧n>1 := Φ[p,n] = Min[Table[
2Φ[p,k]+Φ[p-1, n-k],
{k,1,n-1}]]

Table[Φ[p,n], {n,1,20}, {p,3,10}] // MatrixForm

Table[Φ[p,n], {n,1,100}, {p,4,10}] // MatrixForm

ListPlot[Φ[4,n], {n,1,1000}]

ListPlot[Table[Φ[4,n],{n,1,1000}]]

ListPlot[Table[Log@Φ[4,n],{n,1,10000}]]