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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 Jackson Evoniuk, Steven Klee, Van Magnan, Enumerating Minimal Length Lattice Paths, 2017, also Enumerating Minimal Length Lattice Paths, J. Int. Seq., Vol. 21 (2018), Article 18.3.6. 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 August 18 16:13 EDT 2018. Contains 313833 sequences. (Running on oeis4.)