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

Author

Jon Perry, Nov 24 2012

Keywords

• nonn

Links

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

Crossrefs

Cf. A219670.

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

More terms from Alois P. Heinz, Nov 28 2012