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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)}.

%I A292435 #22 Mar 15 2020 12:05:32
%S A292435 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,
%T A292435 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,
%U A292435 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
%N 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)}.
%H A292435 Jackson Evoniuk, Steven Klee, Van Magnan, <a href="http://fac-staff.seattleu.edu/klees/web/minpaths.pdf">Enumerating Minimal Length Lattice Paths</a>, 2017, also <a href="https://www.emis.de/journals/JIS/VOL21/Klee/klee2.html">Enumerating Minimal Length Lattice Paths</a>, J. Int. Seq., Vol. 21 (2018), Article 18.3.6.
%F A292435 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).
%e A292435 Array T(m,n) begins
%e A292435 n\m    0     1     2     3     4     5     6     7     8     9    10
%e A292435 --------------------------------------------------------------------
%e A292435 [0]    1     1     1     1     2     3     1     3     6     1     4
%e A292435 [1]    1     2     1     4     9     2     9    24     3    16    50
%e A292435 [2]    1     1     4    12     3    15    48     6    36   130    10
%e A292435 [3]    1     4    12     4    21    72    10    64   250    20   150
%e A292435 [4]    2     9     3    21    84    12    88   380    31   255  1215
%e A292435 [5]    3     2    15    72    12    96   460    40   355  1830   101
%e A292435 [6]    1     9    48    10    88   460    44   420  2325   135  1416
%e A292435 [7]    3    24     6    64   380    40   420  2520   155  1740 11046
%e A292435 [8]    6     3    36   250    31   355  2325   155  1860 12600   546
%e A292435 [9]    1    16   130    20   255  1830   135  1740 12600   580  7882
%e A292435 [10]   4    50    10   150  1215   101  1416 11046   546  7882 63056
%o A292435 (Sage)
%o A292435 S = [[1,0], [0,1], [3,0], [2,1], [1,2], [0,3]]
%o A292435 q = 8 # q = range for m,n; change q for more data
%o A292435 numPathsMat = matrix(q+1,q+1,0)
%o A292435 distMatrix  = matrix(q+1,q+1,0)
%o A292435 for m in [0..q]:
%o A292435     for n in [0..q]:
%o A292435         # first determine S-distance to (m,n)
%o A292435         d = minNeighborDist = max(distMatrix.list()) + 1
%o A292435         for s in S:
%o A292435             if m-s[0]>=0 and n-s[1]>=0:
%o A292435                 d = distMatrix[m-s[0],n-s[1]]
%o A292435             if d < minNeighborDist:
%o A292435                 minNeighborDist=d
%o A292435         distMatrix[m,n] = minNeighborDist+1
%o A292435         # next count number of minimal S-paths
%o A292435         count = 0
%o A292435         for s in S:
%o A292435             if m-s[0]>=0 and n-s[1]>=0:
%o A292435                 if distMatrix[m-s[0],n-s[1]]==distMatrix[m,n]-1:
%o A292435                     count += numPathsMat[m-s[0],n-s[1]]
%o A292435         numPathsMat[m,n] = count
%o A292435         numPathsMat[0,0] = 1
%o A292435 print(numPathsMat)
%Y A292435 Cf. A007318.
%K A292435 nonn,tabl,walk
%O A292435 0,5
%A A292435 _Steven Klee_, Dec 08 2017