A346540 -id:A346540 - cp's OEIS frontend

A346541 Number of walks on square lattice from (n,2n) to (0,0) using steps that decrease the Euclidean distance to the origin and increase the Euclidean distance to (n,2n) and that change each coordinate by at most 1.

1, 5, 173, 6273, 327304, 19662204, 1331125733, 97103842536, 7486548949630, 600824064355643, 49716537270181030, 4212436222856773156, 363673201239600512658, 31874623637580787947172, 2828388650238276648013964, 253555200931317108300020394, 22925898959060646660438636660

Offset: 0

Alois P. Heinz, Sep 16 2021

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

Alois P. Heinz, Table of n, a(n) for n = 0..200
Alois P. Heinz, Animation of a(2) = 173 walks
Wikipedia, Counting lattice paths
Wikipedia, Self-avoiding walk

Cf. A346540.

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, 2*n]$2):
seq(a(n), n=0..20);

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, 2n}, {n, 2n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 04 2021, after Alois P. Heinz *)

a(n) = A346540(2n,n) = A346540(n,2n).

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.

Original entry on oeis.org

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

Alois P. Heinz, Sep 20 2021

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

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

Column (or row) k=1 of A346540.

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);

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 *)

cp's OEIS Frontend

A346541 Number of walks on square lattice from (n,2n) to (0,0) using steps that decrease the Euclidean distance to the origin and increase the Euclidean distance to (n,2n) and that change each coordinate by at most 1.

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