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

0,2

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..201

Wikipedia, Counting lattice paths

Wikipedia, Self-avoiding walk

Example

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)).

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)<add(i^2, i=l), b(sort(h)), 0))(l+x), x=s(n))))(nops(l))
    end:
a:= n-> 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 *)

Crossrefs

Column k=3 of A347811.

Cf. A348201.

Sequence in context: A350755 A013764 A078353 * A177818 A269446 A172824

Adjacent sequences: A347810 A347811 A347812 * A347814 A347815 A347816

Keyword

nonn,walk

Author

Alois P. Heinz, Sep 14 2021

Status

approved