VOOZH about

URL: https://oeis.org/A347947

⇱ A347947 - OEIS

login
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.
2
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
0,2
COMMENTS
Lattice points may have negative coordinates, and different walks may differ in length. All walks are self-avoiding.
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=l) and add(i^2, i=v-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<l.l && (v-h).(v-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 *)
CROSSREFS
Column (or row) k=1 of A346540.
Sequence in context: A230985 A286427 A290509 * A208800 A356274 A249935
KEYWORD
nonn,walk
AUTHOR
Alois P. Heinz, Sep 20 2021
STATUS
approved