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], [1, 1]}



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

                                    equation

                          with polynomial coefficients

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

  (m[1] + 1) F(m[1], m[2])   (2 m[2] + 1) F(m[1] + 1, m[2])
- ------------------------ - ------------------------------ + F(2 + m[1], m[2])
          2 + m[1]                      2 + m[1]

     = 0

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

  (m[2] + 1) F(m[1], m[2])   (2 m[1] + 1) F(m[1], m[2] + 1)
- ------------------------ - ------------------------------ + F(m[1], m[2] + 2)
          m[2] + 2                      m[2] + 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], [1, 1]}



            F(n) satisfies the following linear recurrence equation

                          with polynomial coefficients

               (1 + n) F(n)   3 (3 + 2 n) F(1 + n)
               ------------ - -------------------- + F(2 + n) = 0
                  2 + n              2 + n



                       subject to the initial conditions

                         F(0) = 1, F(1) = 3, F(2) = 13

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

                        asymptotically a constant times

                    1/2 n
0.572681633 (3 + 2 2   )

    /               1/2             1/2                  1/2\
    |            3 2      113    9 2        245    1545 2   |
    |    - 1/4 + ------   ---- - ------   - ---- + ---------|
    |              32     1024    128       4096     32768  |   /  1/2
    |1 + -------------- + ------------- + ------------------|  /  n
    |          n                2                  3        | /
    \                          n                  n         /

                           which is roughly equal to

                       n /     0.1174174786   0.01091467142   0.00686523295\
0.572681633 5.828427124  |1. - ------------ + ------------- + -------------|
                         |          n               2               3      |
                         \                         n               n       /

       /  1/2
      /  n
     /



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

[1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719,

    251595969, 1409933619, 7923848253, 44642381823, 252055236609, 1425834724419,

    8079317057869, 45849429914943, 260543813797441, 1482376214227923,

    8443414161166173, 48141245001931263, 274738209148561921,

    1569245074591690083, 8970232353223635949, 51313576749006450879,

    293733710358893793729, 1682471873186160624243, 9642641465118083682429]

               The whole thing took, 2.672, seconds of CPU time