A347947 Number of walks on square lattice from (1,n) to (0,0) using steps that decrease the Euclidean distance to the origin and increase the Euclidean distance to (n,1) and that change each coordinate by at most 1.

1, 3, 5, 24, 81, 298, 1070, 3868, 13960, 50417, 182084, 657707, 2375894, 8583264, 31009890, 112038032, 404803299, 1462624643, 5284813128, 19095564020, 68998567080, 249316670981, 900876831495, 3255230444720, 11762504284218, 42502963168784, 153581776819904

Offset: 0

Author

Alois P. Heinz, Sep 20 2021

Keywords

• nonn
• walk

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..1000
• Alois P. Heinz, Animation of a(5) = 298 walks
• Wikipedia, Counting lattice paths
• Wikipedia, Self-avoiding walk

Crossrefs

Column (or row) k=1 of A346540.

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, v) option remember; (n-> `if`(l=[0$n], 1, add((h-> `if`(
      add(i^2, i=h)add(i^2, i=v-l)
      , b(h, v), 0))(l+x), x=s(n))))(nops(l))
    end:
a:= n-> b([n, 1]$2):
seq(a(n), n=0..30);

Mathematica

s[n_] := s[n] = If[n == 0, {{}}, Sequence @@
     Table[Append[#, i], {i, -1, 1}]& /@ s[n-1]];
b[l_, v_] := b[l, v] = With[{n = Length[l]},
     If[l == Table[0, {n}], 1, Sum[With[{h = l + x},
     If[h.h(v-l).(v-l), b[h, v], 0]], {x, s[n]}]]];
a[n_] := b[{n, 1}, {n, 1}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 04 2021, after Alois P. Heinz *)