Mean geodesic distance of the level-\(n\) Sierpinski gasket (Q2062592)

It is known that the mean geodesic distance of the classical Sierpinski gasket is \(466/885\), see [\textit{A. M. Hinz} and \textit{A. Schief}, Probab. Theory Relat. Fields 87, No. 1, 129--138 (1990; Zbl 0688.60074)]. Here a (quite complicated) formula for the mean geodesic distance of the level-n Sierpinski gasket is derived. For instance we get \(21419/43248\) for \(n=3\) and \(676659/1421390\) for \(n=4\), the limit \(n\) to \(\infty\) is \(2/5\). To prove the result the integral of geodesic distance with respect to self-similar measure is studied by means of the finite pattern phenomenon of integrals. The paper is well written and illustrated.