A328299 -id:A328299 - cp's OEIS frontend

A328297 Number T(n,k) of n-step walks on cubic lattice starting at (0,0,0), ending at (x,y,z) with x=k, remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 2, 1, 5, 6, 1, 16, 26, 14, 1, 58, 112, 93, 30, 1, 228, 489, 522, 288, 62, 1, 945, 2182, 2737, 2040, 825, 126, 1, 4072, 9934, 13934, 12642, 7210, 2254, 254, 1, 18078, 46016, 70058, 72994, 52086, 23878, 5969, 510, 1, 82172, 216322, 350648, 404788, 338520, 198795, 75570, 15468, 1022, 1

Offset: 0

Alois P. Heinz, Oct 11 2019

			Triangle T(n,k) begins:
     1;
     2,    1;
     5,    6,     1;
    16,   26,    14,     1;
    58,  112,    93,    30,    1;
   228,  489,   522,   288,   62,    1;
   945, 2182,  2737,  2040,  825,  126,   1;
  4072, 9934, 13934, 12642, 7210, 2254, 254, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Column k=0 gives A328296.
Main diagonal gives A000012.
T(n,n-1) gives A000918(n+1).
T(2n,n) gives A328427.
Row sums give A328295.
Cf. A038207, A328299, A328300.

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
      add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
      sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
    end:
T:= (n, k)-> add(b(sort([k, j, n-k-j])), j=0..n-k):
seq(seq(T(n, k), k=0..n), n=0..12);

b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
T[n_, k_] := Sum[b[Sort[{k, j, n - k - j}]], {j, 0, n - k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 12 2020, after Maple *)

A328345 Number of n-step walks on cubic lattice starting at (0,0,0), ending at (floor(n/3), floor((n+1)/3), floor((n+2)/3)) and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).

Original entry on oeis.org

1, 1, 4, 21, 88, 440, 2385, 11781, 62832, 352128, 1842240, 10132320, 57775905, 311810785, 1746140396, 10060071021, 55367204256, 313747490784, 1820016119376, 10152658848528, 58015193420160, 338183208699840, 1905152077559808, 10954624445968896, 64089909936535329

Offset: 0

Alois P. Heinz, Oct 13 2019

These walks are not restricted to the first (nonnegative) octant.

			a(2) = 4: [(0,0,0),(1,0,0),(0,1,1)], [(0,0,0),(0,1,0),(0,1,1)], [(0,0,0),(0,0,1),(0,1,1)], [(0,0,0),(-1,1,1),(0,1,1)].

Alois P. Heinz, Table of n, a(n) for n = 0..650
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Cf. A002426, A328299, A328347.

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(add(
      add(`if`(i+j+k=1, (h-> `if`(add(t, t=h)<0, 0, b(h)))(
      sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
    end:
a:= n-> b([floor((n+i)/3)$i=0..2]):
seq(a(n), n=0..24);

b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[Total[h] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
a[n_] := b[Table[Floor[(n + i)/3], {i, 0, 2}]];
a /@ Range[0, 24] (* Jean-François Alcover, May 12 2020, after Maple *)

cp's OEIS Frontend

A328297 Number T(n,k) of n-step walks on cubic lattice starting at (0,0,0), ending at (x,y,z) with x=k, remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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