A347811 -id:A347811 - cp's OEIS frontend

A346539 a(n) is the number of paths in the Z X Z grid joining (0,0) and (n,n) each of whose steps increases the Euclidean distance to the origin and has coordinates with absolute value at most 1.

1, 3, 25, 241, 2545, 28203, 322681, 3776275, 44947503, 542097295, 6607714859, 81247609095, 1006335719467, 12542292874825, 157159924565801, 1978517963096763, 25010881408459855, 317327992746937599, 4039340709637022007, 51569571332132589961, 660140626022179390983

Offset: 0

José María Grau Ribas, Sep 10 2021

nonn

All terms are odd.

Alois P. Heinz, Table of n, a(n) for n = 0..886
Alois P. Heinz, Animation of a(3) = 241 paths

Main diagonal of A346538.
Column k=2 of A347811.
Cf. A008288, A001850, A225042.

b:= proc(n, k) option remember; `if`([n, k]=[0$2], 1, add(add(
     `if`(i^2+j^2 b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 12 2021

rodean[{m_, n_}] := Select[ Complement[ Flatten[Table[{m, n} + {s, t}, {s, -1, 1}, {t, -1, 1}], 1] // Union, {{m, n}}], #[[1]]^2 + #[[2]]^2 < m^2 + n^2 &];
$RecursionLimit=10^6; Clear[T]; T[{0, 0}]=1; T[{m_,n_}]:= T[{m,n}]= Sum[T[rodean[{m,n}][[i]]],{i,Length[rodean[{m, n}]]}]; Table[T[{n,n}],{n, 0,22}]
(* Second program: *)
b[n_, k_] := b[n, k] = If[{n, k} == {0, 0}, 1, Sum[Sum[If[i^2 + j^2 < n^2 + k^2, b@@Sort[{i, j}], 0], {j, k-1, k+1}], {i, n-1, n+1}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 03 2021, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 1/6*(19009+153*sqrt(17))^(1/3) + 356/(3*(19009+153*sqrt(17))^(1/3)) + 14/3 = 13.56165398271839628518... and c = 2.3842296614800994817864695565477260682981556338086519... . - Vaclav Kotesovec, Sep 13 2021

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.

Original entry on oeis.org

1, 19, 211075, 2062017739, 32191353922714, 977270269148852086, 29618256217540107753856, 1041952262234097478667071246, 43960391382107369608617444946360, 2007170356703297211447385988052335644, 99624394337129260265907069889802324849302

Offset: 0

Alois P. Heinz, Sep 14 2021

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

			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)).

Alois P. Heinz, Table of n, a(n) for n = 0..201
Wikipedia, Counting lattice paths
Wikipedia, Self-avoiding walk

Column k=3 of A347811.

Cf. A348201.

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..12);

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 *)

A347810 Number of n-dimensional lattice walks from {n}^n to {0}^n using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1.

Original entry on oeis.org

1, 1, 25, 2062017739, 255053951339165796439851848897794625

Offset: 0

Alois P. Heinz, Sep 14 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..6
Alois P. Heinz, Animation of a(2) = 25 walks
Wikipedia, Counting lattice paths
Wikipedia, Self-avoiding walk

Main diagonal of A347811.

Cf. A034841.

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$n], 1, add((h-> `if`(
      add(i^2, i=h) b([n$n]):
seq(a(n), n=0..5);

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[Table[n, {n}]];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Nov 04 2021, after Alois P. Heinz *)

A347812 Number of n-dimensional lattice walks from {2}^n to {0}^n using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1.

Original entry on oeis.org

1, 1, 25, 211075, 1322634996717, 16042961630858858915656, 286729345864079773218271997053157611, 25868451537111690721940670963124809063875212336403319, 3742158706432626794575922563227094346392414743343045621639247710036163317

Offset: 0

Alois P. Heinz, Sep 14 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..10
Alois P. Heinz, Animation of a(2) = 25 walks
Wikipedia, Counting lattice paths
Wikipedia, Self-avoiding walk

Row n=2 of A347811.

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$n], 1, add((h-> `if`(
      add(i^2, i=h) b([2$n]):
seq(a(n), n=0..7);

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[Table[2, {n}]];
Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Nov 04 2021, after Alois P. Heinz *)

A346840 Number of n-dimensional lattice walks from {1}^n to {0}^n using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1.

Original entry on oeis.org

1, 1, 3, 19, 323, 38716, 32253681, 78267222216, 762698791293515, 37603168183752885658, 9243338412512497704718998, 11335247475355582487279558180986, 68305298969167998414438069494886302081, 1920885596256995709122789811280666218400361901

Offset: 0

Alois P. Heinz, Sep 14 2021

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

			a(3) = 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)).

Wikipedia, Counting lattice paths
Wikipedia, Self-avoiding walk

Row n=1 of A347811.

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$n], 1, add((h-> `if`(
      add(i^2, i=h) b([1$n]):
seq(a(n), n=0..10);

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[Table[1, {n}]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 04 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A346539 a(n) is the number of paths in the Z X Z grid joining (0,0) and (n,n) each of whose steps increases the Euclidean distance to the origin and has coordinates with absolute value at most 1.

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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.

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A347810 Number of n-dimensional lattice walks from {n}^n to {0}^n using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1.

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Maple

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A347812 Number of n-dimensional lattice walks from {2}^n to {0}^n using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1.

Views

Author

Keywords

Comments

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Crossrefs

Programs

Maple

Mathematica

A346840 Number of n-dimensional lattice walks from {1}^n to {0}^n using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica