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

0,3

Links

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

Formula

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

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 *)

Crossrefs

Cf. A219670.

Sequence in context: A207995 A008788 A360759 * A138460 A182148 A304170

Adjacent sequences: A219668 A219669 A219670 * A219672 A219673 A219674

Keyword

nonn

Author

Jon Perry, Nov 24 2012

Extensions

More terms from Alois P. Heinz, Nov 28 2012

Status

approved