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ways are there to move from (0, 0) to (n, 2), under lattice point conditions

Source: 41st Austrian Mathematical Olympiad National Competition (Final Round, part 2) 3rd June 2010 p4

September 5, 2019

combinatoricscombinatorial geometrylattice points

Problem Statement

Consider the part of a lattice given by the corners

(0, 0), (n, 0), (n, 2)

and

(0, 2)

. From a lattice point

(a, b)

one can move to

(a + 1, b)

or to

(a + 1, b + 1)

or to

(a, b - 1

), provided that the second point is also contained in the part of the lattice. How many ways are there to move from

(0, 0)

(n, 2)

considering these rules?