A347813 - OEIS

%I #19 Nov 04 2021 05:58:09

%S 1,19,211075,2062017739,32191353922714,977270269148852086,

%T 29618256217540107753856,1041952262234097478667071246,

%U 43960391382107369608617444946360,2007170356703297211447385988052335644,99624394337129260265907069889802324849302

%N 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.

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

%H Alois P. Heinz, <a href="/A347813/b347813.txt">Table of n, a(n) for n = 0..201</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>

%e a(1) = 19:

%e   ((1,1,1), (0,0,0)),

%e   ((1,1,1), (0,0,1), (0,0,0)),

%e   ((1,1,1), (0,1,0), (0,0,0)),

%e   ((1,1,1), (0,1,1), (0,0,0)),

%e   ((1,1,1), (1,0,0), (0,0,0)),

%e   ((1,1,1), (1,0,1), (0,0,0)),

%e   ((1,1,1), (1,1,0), (0,0,0)),

%e   ((1,1,1), (0,1,1), (-1,0,0), (0,0,0)),

%e   ((1,1,1), (0,1,1), (0,0,1), (0,0,0)),

%e   ((1,1,1), (0,1,1), (0,1,0), (0,0,0)),

%e   ((1,1,1), (0,1,1), (1,0,0), (0,0,0)),

%e   ((1,1,1), (1,0,1), (0,-1,0), (0,0,0)),

%e   ((1,1,1), (1,0,1), (0,0,1), (0,0,0)),

%e   ((1,1,1), (1,0,1), (0,1,0), (0,0,0)),

%e   ((1,1,1), (1,0,1), (1,0,0), (0,0,0)),

%e   ((1,1,1), (1,1,0), (0,0,-1), (0,0,0)),

%e   ((1,1,1), (1,1,0), (0,0,1), (0,0,0)),

%e   ((1,1,1), (1,1,0), (0,1,0), (0,0,0)),

%e   ((1,1,1), (1,1,0), (1,0,0), (0,0,0)).

%p s:= proc(n) option remember;

%p      `if`(n=0, [[]], map(x-> seq([x[], i], i=-1..1), s(n-1)))

%p     end:

%p b:= proc(l) option remember; (n-> `if`(l=[0$n], 1, add((h-> `if`(

%p       add(i^2, i=h)<add(i^2, i=l), b(sort(h)), 0))(l+x), x=s(n))))(nops(l))

%p     end:

%p a:= n-> b([n$3]):

%p seq(a(n), n=0..12);

%t s[n_] := s[n] = If[n == 0, {{}}, Sequence @@ Table[Append[#, i], {i, -1, 1}]& /@ s[n-1]];

%t 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]}]]];

%t a[n_] := b[{n, n, n}];

%t Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Nov 04 2021, after _Alois P. Heinz_ *)

%Y Column k=3 of A347811.

%Y Cf. A348201.

%K nonn,walk

%O 0,2

%A _Alois P. Heinz_, Sep 14 2021