The Moser's formula for the division of the circle by chords problem revisited

The enumeration of the regions formed when circle is divided by secants drawn from points on the circle is one of the examples where the inductive reasoning fails as was pointed out by Leo Moser in the Mathematical Miscellany in 1949. The formula that gives the right number of regions can be deduced by combinatorics reasoning using the Euler’s planar graph formula, etc. My contribution in the present work is to reformulate and solve such problem in terms of a fourth order difference equation and to obtain the formula proposed by Leo Moser.