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A292435 Array T read by antidiagonals: T(m,n) = number of lattice walks of minimal length from (0,0) to (m,n) using steps in directions from {(1,0), (0,1), (3,0), (2,1), (1,2), (0,3)}. 0
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 4, 4, 4, 2, 3, 9, 12, 12, 9, 3, 1, 2, 3, 4, 3, 2, 1, 3, 9, 15, 21, 21, 15, 9, 3, 6, 24, 48, 72, 84, 72, 48, 24, 6, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 4, 16, 36, 64, 88, 96, 88, 64, 36, 16, 4, 10, 50, 130, 250, 380, 460, 460, 380, 250, 130, 50, 10, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..90.

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

FORMULA

G.f.: Sum(T(m,n)*x^m*y^n,m>=0,n>=0) = Sum(binomial(q+r,r)*(x^3+x^2*y+x*y^2+y^3)^q*(x+y)^r,q>=0,0<=r<=2).

EXAMPLE

Array T(m,n) begins

n\m  0     1     2     3     4     5     6     7     8     9    10

0    1     1     1     1     2     3     1     3     6     1     4

1    1     2     1     4     9     2     9    24     3    16    50

2    1     1     4    12     3    15    48     6    36   130    10

3    1     4    12     4    21    72    10    64   250    20   150

4    2     9     3    21    84    12    88   380    31   255  1215

5    3     2    15    72    12    96   460    40   355  1830   101

6    1     9    48    10    88   460    44   420  2325   135  1416

7    3    24     6    64   380    40   420  2520   155  1740 11046

8    6     3    36   250    31   355  2325   155  1860 12600   546

9    1    16   130    20   255  1830   135  1740 12600   580  7882

10   4    50    10   150  1215   101  1416 11046   546  7882 63056

PROG

(Sage)

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

q = 8 # q = range for m, n; change q for more data

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

distMatrix  = matrix(q+1, q+1, 0)

for m in [0..q]:

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

........# first determine S-distance to (m, n)

........minNeighborDist = max(distMatrix.list()) + 1

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

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

................d = distMatrix[m-s[0], n-s[1]]

............if d < minNeighborDist:

................minNeighborDist=d

........distMatrix[m, n] = minNeighborDist+1

........# next count number of minimal S-paths

........count = 0

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

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

................if distMatrix[m-s[0], n-s[1]]==distMatrix[m, n]-1:

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

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

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

print numPathsMat

CROSSREFS

Cf. A007318.

Sequence in context: A046214 A232088 A115413 * A069283 A285337 A033630

Adjacent sequences:  A292432 A292433 A292434 * A292436 A292437 A292438

KEYWORD

nonn,tabl,walk

AUTHOR

Steven Klee, Dec 08 2017

STATUS

approved

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Last modified June 19 08:25 EDT 2018. Contains 305581 sequences. (Running on oeis4.)