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 A292436 Array T read by antidiagonals: T(m,n) is the number of lattice walks of minimal length from (0,0) to (m,n) using steps in directions from {(1,0), (0,1), (2,1), (1,2)}. 0
 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 9, 9, 3, 1, 1, 4, 1, 2, 1, 4, 1, 1, 5, 3, 9, 9, 3, 5, 1, 1, 6, 6, 24, 36, 24, 6, 6, 1, 1, 7, 10, 1, 3, 3, 1, 10, 7, 1, 1, 8, 15, 4, 16, 24, 16, 4, 15, 8, 1, 1, 9, 21, 10, 50, 100, 100, 50, 10, 21, 9, 1, 1, 10, 28, 20, 1, 4, 6, 4, 1, 20, 28, 10, 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 T(m,n) = binomial(m-n,n) for m>=2*n; T(m,n) = binomial(q+r,r)*binomial(q+r,m-q) for 1/2*n<=m<=2*n, where m+n = 3*q+r with 0<=r<=2; T(m,n) = binomial(n-m,m) for m<=1/2*n. EXAMPLE Array T(m,n) begins n\m 0    1    2    3    4    5    6    7    8    9   10 0   1    1    1    1    1    1    1    1    1    1    1 1   1    2    1    2    3    4    5    6    7    8    9 2   1    1    4    9    1    3    6   10   15   21   28 3   1    2    9    2    9   24    1    4   10   20   35 4   1    3    1    9   36    3   16   50    1    5   15 5   1    4    3   24    3   24  100    4   25   90    1 6   1    5    6    1   16  100    6   50  225    5   36 7   1    6   10    4   50    4   50  300   10   90  441 8   1    7   15   10    1   25  225   10  120  735   15 9   1    8   21   20    5   90    5   90  735   20  245 10  1    9   28   35   15    1   36  441   15  245 1960 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: A175466 A214403 A261527 * A184097 A205399 A135303 Adjacent sequences:  A292433 A292434 A292435 * A292437 A292438 A292439 KEYWORD nonn,walk,tabl AUTHOR Steven Klee, Dec 08 2017 STATUS approved

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Last modified January 18 23:05 EST 2019. Contains 319282 sequences. (Running on oeis4.)