``
                Let, F([m[1], m[2]]),  be the number of ways of

                                of walking from

                                     [0, 0]

                        to the point , m = [m[1], m[2]]

                    in the , 2, -dimensional cubic lattice

                       using the following allowed steps:

                        {[0, 1], [1, 0], [0, 2], [2, 0]}



           F(m[1], m[2]), satisfies the following linear recurrences

                                    equation

                          with polynomial coefficients

                     in the, m[1], direction, it satisfies

    (m[2] + m[1] + 3) (m[2] + m[1] + 2) F(m[1], m[2])
4/5 ------------------------------------------------- + 2/5
                  (2 + m[1]) (m[1] + 4)

                                       2
    (2 m[1] m[2] + 5 m[2] + 26 + 4 m[1]  + 21 m[1]) (m[2] + m[1] + 3)

    F(m[1] + 1, m[2])/((m[1] + 4) (m[1] + 3) (2 + m[1])) + 1/5

            2                                              2
    (-5 m[1]  - 33 m[1] - 6 m[1] m[2] - 17 m[2] - 54 + m[2] ) F(2 + m[1], m[2])

                                   (9 m[1] + 5 m[2] + 34) F(m[1] + 3, m[2])
    /((m[1] + 4) (m[1] + 3)) - 1/5 ----------------------------------------
                                                   m[1] + 4

     + F(m[1] + 4, m[2]) = 0

                     in the, m[2], direction, it satisfies

    (m[2] + m[1] + 3) (m[2] + m[1] + 2) F(m[1], m[2])
4/5 ------------------------------------------------- + 2/5
                  (2 + m[2]) (4 + m[2])

                                                 2
    (5 m[1] + 2 m[1] m[2] + 26 + 21 m[2] + 4 m[2] ) (m[2] + m[1] + 3)

    F(m[1], 1 + m[2])/((4 + m[2]) (3 + m[2]) (2 + m[2])) - 1/5

           2                                              2
    (5 m[2]  + 33 m[2] + 6 m[1] m[2] + 17 m[1] + 54 - m[1] ) F(m[1], 2 + m[2])/

                                  (9 m[2] + 34 + 5 m[1]) F(m[1], 3 + m[2])
    ((3 + m[2]) (4 + m[2])) - 1/5 ----------------------------------------
                                                  4 + m[2]

     + F(m[1], 4 + m[2]) = 0

        Let F(n) be the number of ways of walking from , [0, 0], to the

            point , [n, n], in the , 2, -dimensional cubic lattice

                       using the following allowed steps:

                        {[0, 1], [1, 0], [0, 2], [2, 0]}



            F(n) satisfies the following linear recurrence equation

                          with polynomial coefficients

     (2 n + 3) (11 n + 26) (n + 1) F(n)
16/5 ----------------------------------
        (n + 2) (11 n + 15) (n + 3)

                 3        2
           (121 n  + 649 n  + 1135 n + 646) F(n + 1)
     - 4/5 -----------------------------------------
                  (n + 2) (11 n + 15) (n + 3)

                 2
           (176 n  + 680 n + 605) F(n + 2)
     - 2/5 ------------------------------- + F(n + 3) = 0
                 (n + 3) (11 n + 15)



                       subject to the initial conditions

                    F(0) = 1, F(1) = 2, F(2) = 14, F(3) = 84

            This implies, thanks to Birkhoff-Trijinski, that F(n) is

                        asymptotically a constant times

                   1/2 n
0.37305616 (4 + 2 3   )

    /        1/2                        1/2            1/2           \
    |    67 3      119    6253    7163 3        32645 3       129625 |
    |    ------- - ---   ------ - ---------   - ---------- + --------|
    |     1452     484   117128    234256        15460896    10307264|   /  1/2
    |1 + ------------- + ------------------ + -----------------------|  /  n
    |          n                  2                      3           | /
    \                            n                      n            /

                           which is roughly equal to

                      n /     0.1659453140   0.00042398086   0.008918933381\
0.37305616 7.464101616  |1. - ------------ + ------------- + --------------|
                        |          n               2                3      |
                        \                         n                n       /

       /  1/2
      /  n
     /



             For the record, the first 31 terms of the sequence are

[1, 2, 14, 84, 556, 3736, 25612, 177688, 1244398, 8777612, 62271384, 443847648,

    3175924636, 22799963576, 164142004184, 1184574592592, 8567000931404,

    62073936511496, 450518481039956, 3274628801768744, 23833760489660324,

    173679413875623368, 1267013689048017584, 9252299435205985664,

    67626504432024377756, 494710324956646794296, 3621791112234327295592,

    26534383313499716907504, 194529413506239838951024,

    1427026630616364232856416, 10474450985957996054150576]

               The whole thing took, 10.060, seconds of CPU time