A347813 Number of cubic lattice walks from (n,n,n) to (0,0,0) using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1.

1, 19, 211075, 2062017739, 32191353922714, 977270269148852086, 29618256217540107753856, 1041952262234097478667071246, 43960391382107369608617444946360, 2007170356703297211447385988052335644, 99624394337129260265907069889802324849302

Offset: 0

Author

Alois P. Heinz, Sep 14 2021

Keywords

• nonn
• walk

Comments

Lattice points may have negative coordinates, and different walks may differ in length. All walks are self-avoiding.

Examples

			a(1) = 19:
  ((1,1,1), (0,0,0)),
  ((1,1,1), (0,0,1), (0,0,0)),
  ((1,1,1), (0,1,0), (0,0,0)),
  ((1,1,1), (0,1,1), (0,0,0)),
  ((1,1,1), (1,0,0), (0,0,0)),
  ((1,1,1), (1,0,1), (0,0,0)),
  ((1,1,1), (1,1,0), (0,0,0)),
  ((1,1,1), (0,1,1), (-1,0,0), (0,0,0)),
  ((1,1,1), (0,1,1), (0,0,1), (0,0,0)),
  ((1,1,1), (0,1,1), (0,1,0), (0,0,0)),
  ((1,1,1), (0,1,1), (1,0,0), (0,0,0)),
  ((1,1,1), (1,0,1), (0,-1,0), (0,0,0)),
  ((1,1,1), (1,0,1), (0,0,1), (0,0,0)),
  ((1,1,1), (1,0,1), (0,1,0), (0,0,0)),
  ((1,1,1), (1,0,1), (1,0,0), (0,0,0)),
  ((1,1,1), (1,1,0), (0,0,-1), (0,0,0)),
  ((1,1,1), (1,1,0), (0,0,1), (0,0,0)),
  ((1,1,1), (1,1,0), (0,1,0), (0,0,0)),
  ((1,1,1), (1,1,0), (1,0,0), (0,0,0)).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..201
• Wikipedia, Counting lattice paths
• Wikipedia, Self-avoiding walk

Crossrefs

Column k=3 of A347811.

Cf. A348201.

Programs

Maple

s:= proc(n) option remember;
     `if`(n=0, [[]], map(x-> seq([x[], i], i=-1..1), s(n-1)))
    end:
b:= proc(l) option remember; (n-> `if`(l=[0$n], 1, add((h-> `if`(
      add(i^2, i=h) b([n$3]):
seq(a(n), n=0..12);

Mathematica

s[n_] := s[n] = If[n == 0, {{}}, Sequence @@ Table[Append[#, i], {i, -1, 1}]& /@ s[n-1]];
b[l_List] := b[l] = With[{n = Length[l]}, If[l == Table[0, {n}], 1, Sum[With[{h = l+x}, If[h.h < l.l, b[Sort[h]], 0]], {x, s[n]}]]];
a[n_] := b[{n, n, n}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 04 2021, after Alois P. Heinz *)