Authors: Jonathan M. Borwein ^1,2; Dirk Nuyens ^3,4; Armin Straub ⁵; James Wan ^1,2

• 1 University of Newcastle [Australia]
• 2 University of Newcastle [Callaghan, Australia]
• 3 Catholic University of Leuven - Katholieke Universiteit Leuven [KU Leuven]
• 4 Catholic University of Leuven = Katholieke Universiteit Leuven [KU Leuven]
• 5 Tulane University

We study the expected distance of a two-dimensional walk in the plane with unit steps in random directions. A series evaluation and recursions are obtained making it possible to explicitly formulate this distance for small number of steps. Formulae for all the moments of a 2-step and a 3-step walk are given, and an expression is conjectured for the 4-step walk. The paper makes use of the combinatorical features exhibited by the even moments which, for instance, lead to analytic continuations of the underlying integral.