A348201 Number of walks on cubic lattice from (n,n,n) to (0,0,0) using steps that decrease the Euclidean distance to the origin and that change each coordinate by 1 or by -1.

1, 1, 25, 211, 4057, 79945, 1559719, 34166335, 784027759, 18367309153, 447879467629, 11160419719795, 283032843838285, 7307188685246689, 191139484940529781, 5056715112537049897, 135152031778121985907, 3642958379395296513337, 98930628058690700138443

Offset: 0

Author

Alois P. Heinz, Oct 06 2021

Keywords

• nonn
• walk

Comments

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

All terms are odd.

Examples

			a(2) = 25:
  ((2,2,2), (1,1,1), (0,0,0)),
  ((2,2,2), (1,1,3), (0,0,2), (-1,-1,1), (0,0,0)),
  ((2,2,2), (1,1,3), (0,0,2), (-1,1,1), (0,0,0)),
  ((2,2,2), (1,1,3), (0,0,2), (1,-1,1), (0,0,0)),
  ((2,2,2), (1,1,3), (0,0,2), (1,1,1), (0,0,0)),
  ((2,2,2), (1,1,3), (0,2,2), (-1,1,1), (0,0,0)),
  ((2,2,2), (1,1,3), (0,2,2), (1,1,1), (0,0,0)),
  ((2,2,2), (1,1,3), (2,0,2), (1,-1,1), (0,0,0)),
  ((2,2,2), (1,1,3), (2,0,2), (1,1,1), (0,0,0)),
  ((2,2,2), (1,3,1), (0,2,0), (-1,1,-1), (0,0,0)),
  ((2,2,2), (1,3,1), (0,2,0), (-1,1,1), (0,0,0)),
  ((2,2,2), (1,3,1), (0,2,0), (1,1,-1), (0,0,0)),
  ((2,2,2), (1,3,1), (0,2,0), (1,1,1), (0,0,0)),
  ((2,2,2), (1,3,1), (0,2,2), (-1,1,1), (0,0,0)),
  ((2,2,2), (1,3,1), (0,2,2), (1,1,1), (0,0,0)),
  ((2,2,2), (1,3,1), (2,2,0), (1,1,-1), (0,0,0)),
  ((2,2,2), (1,3,1), (2,2,0), (1,1,1), (0,0,0)),
  ((2,2,2), (3,1,1), (2,0,0), (1,-1,-1), (0,0,0)),
  ((2,2,2), (3,1,1), (2,0,0), (1,-1,1), (0,0,0)),
  ((2,2,2), (3,1,1), (2,0,0), (1,1,-1), (0,0,0)),
  ((2,2,2), (3,1,1), (2,0,0), (1,1,1), (0,0,0)),
  ((2,2,2), (3,1,1), (2,0,2), (1,-1,1), (0,0,0)),
  ((2,2,2), (3,1,1), (2,0,2), (1,1,1), (0,0,0)),
  ((2,2,2), (3,1,1), (2,2,0), (1,1,-1), (0,0,0)),
  ((2,2,2), (3,1,1), (2,2,0), (1,1,1), (0,0,0)).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..681
• Wikipedia, Counting lattice paths
• Wikipedia, Self-avoiding walk

Crossrefs

Cf. A347813.

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..20);

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, 20}] (* Jean-François Alcover, May 23 2024, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n, where d = (3*(292 + 4*sqrt(5))^(1/3))/2 + 66/(292 + 4*sqrt(5))^(1/3) + 10 = 29.900786688498085577218938127572448... and c = 0.00221301854906444252905280527969234142... - Vaclav Kotesovec, Oct 24 2021