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A292437 a(n) is the number of lattice walks from (0,0) to (3*n,3*n) that use steps in directions {(3,0), (2,1), (1,2), (0,3)} and stay weakly below the line y=x. 1
1, 2, 13, 120, 1288, 15046, 185658, 2380720, 31411376, 423660504, 5814905977, 80956085304, 1140478875656, 16227516683124, 232870988052180, 3366482778363616, 48981220255732960, 716707681487535144, 10539913681632290532, 155697664218428455520, 2309297999296926348448 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..834

J. Evoniuk, S. Klee, and V. Magnan, Enumerating Minimal Length Lattice Paths, submitted, 2017.

EXAMPLE

For n=2, the a(2)=13 paths terminating at (6,6) are

(3, 0), (3, 0), (0, 3), (0, 3)

(3, 0), (2, 1), (1, 2), (0, 3)

(3, 0), (2, 1), (0, 3), (1, 2)

(3, 0), (1, 2), (2, 1), (0, 3)

(3, 0), (1, 2), (1, 2), (1, 2)

(3, 0), (0, 3), (3, 0), (0, 3)

(3, 0), (0, 3), (2, 1), (1, 2)

(2, 1), (3, 0), (1, 2), (0, 3)

(2, 1), (3, 0), (0, 3), (1, 2)

(2, 1), (2, 1), (2, 1), (0, 3)

(2, 1), (2, 1), (1, 2), (1, 2)

(2, 1), (1, 2), (3, 0), (0, 3)

(2, 1), (1, 2), (2, 1), (1, 2)

MAPLE

b:= proc(l) option remember; `if`(l=[0$2], 1, add(

      (f-> `if`(min(f)<0 or f[1]<f[2], 0, b(f)))(l-g),

       g=[[3, 0], [2, 1], [1, 2], [0, 3]]))

    end:

a:= n-> b([3*n$2]):

seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2017

PROG

(Sage)

S = [[3, 0], [2, 1], [1, 2], [0, 3]]

q = 10

numPathsMat = matrix(q+1, q+1, 0)

for m in [0..q]:

....for n in [0..m]:

........count = 0

........for s in S:

............if n-s[1]>=0 and m-s[0]>=n-s[1]:

................count += numPathsMat[m-s[0], n-s[1]]

........numPathsMat[m, n] = count

........numPathsMat[0, 0] = 1

print numPathsMat.diagonal()

CROSSREFS

Cf. A000108, A007318.

Sequence in context: A209217 A000180 A215715 * A192460 A004122 A086630

Adjacent sequences:  A292434 A292435 A292436 * A292439 A292440 A292441

KEYWORD

nonn,walk

AUTHOR

Steven Klee, Dec 08 2017

STATUS

approved

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Last modified February 20 18:57 EST 2018. Contains 299381 sequences. (Running on oeis4.)