A219670 -id:A219670 - cp's OEIS frontend

A219671 Number of n-step paths on cubic lattice from (0,0,0) to (1,0,0) with moves in any direction but no zero moves allowed.

0, 1, 16, 243, 4704, 90930, 1883760, 39868955, 867923840, 19226700486, 432776971200, 9863289713046, 227212909995456, 5281459355486028, 123725917334379360, 2918138849807324715, 69236356202861088384, 1651381196044566423294, 39572852284708565895072

Offset: 0

Jon Perry, Nov 24 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A219670.

b:= proc(n,p) option remember; `if`(p[3]>n, 0, `if`(n=0, 1,
      add(add(add(`if`(i=0 and j=0 and k=0, 0, b(n-1, sort(map(abs,
      p+[i, j, k])))), i=-1..1), j=-1..1), k=-1..1)))
    end:
a:= n-> b(n, [0$2, 1]):
seq (a(n), n=0..25);  # Alois P. Heinz, Nov 28 2012

b[n_, p_] := b[n, p] = If[p[[3]]>n, 0, If[n==0, 1, Sum[Sum[Sum[If[i==0 && j==0 && k==0, 0, b[n-1, Sort[Map[Abs, p + {i, j, k}]]]], {i, -1, 1}], {j, -1, 1}], {k, -1, 1}]]];
a[n_] := b[n, {0, 0, 1}];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) ~ c * 26^n / n^(3/2), where c = 0.1102253437... . - Vaclav Kotesovec, Sep 07 2014

More terms from Alois P. Heinz, Nov 28 2012

A219986 Number of n-step paths on a quartic lattice that move from (0,0,0,0) to (1,0,0,1) allowing all moves in {-1,0,1}^4 inclusive the zero move.

Original entry on oeis.org

0, 1, 36, 1764, 92416, 5267025, 315630756, 19684370601, 1264977082944, 83266957255329, 5588691282002500, 381203015928291216, 26357375491548319296, 1843677173726039815969, 130261796682232750056900, 9284172482167489217304900, 666818520819487582805692416

Offset: 0

Jon Perry, Dec 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60

Cf. A219670, A219671.

b=new Array();
for (i1=-1;i1<2;i1++)
for (i2=-1;i2<2;i2++)
for (i3=-1;i3<2;i3++)
for (i4=-1;i4<2;i4++) {
n=(i1+1)+(i2+1)*3+(i3+1)*9+(i4+1)*27;
b[n]=[i1,i2,i3,i4];
}
function inc(arr,m) {
al=arr.length-1;
full=true;
for (ac=0;ac<=al;ac++) if (arr[ac]!=m) {full=false;break;}
if (full==true) return false;
while (arr[al]==m && al>0) {arr[al]=0;al--;}
arr[al]++;
return true;
}
for (k=0;k<5;k++) {
c=0;
a=new Array();
for (i=0;i

b:= proc(n, p) option remember; `if`(p[4]>n, 0, `if`(n=0, 1,
      add(add(add(add(b(n-1, sort(map(abs, p+[i, j, k, m])))
      , i=-1..1), j=-1..1), k=-1..1), m=-1..1)))
    end:
a:= n-> b(n, [0$2, 1$2]):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 02 2012

b[n_, p_] := b[n, p] = If[p[[4]] > n, 0, If[n == 0, 1,
     Sum[Sum[Sum[Sum[b[n-1, Sort[Abs[ p + {i, j, k, m}]]],
     {i, -1, 1}], {j, -1, 1}], {k, -1, 1}], {m, -1, 1}]]];
a[n_] := b[n, {0, 0, 1, 1}];
Table [a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)

More terms from Alois P. Heinz, Dec 02 2012

cp's OEIS Frontend

A219671 Number of n-step paths on cubic lattice from (0,0,0) to (1,0,0) with moves in any direction but no zero moves allowed.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions