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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.

%I A347813 #19 Nov 04 2021 05:58:09
%S A347813 1,19,211075,2062017739,32191353922714,977270269148852086,
%T A347813 29618256217540107753856,1041952262234097478667071246,
%U A347813 43960391382107369608617444946360,2007170356703297211447385988052335644,99624394337129260265907069889802324849302
%N 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.
%C A347813 Lattice points may have negative coordinates, and different walks may differ in length.  All walks are self-avoiding.
%H A347813 Alois P. Heinz, <a href="/A347813/b347813.txt">Table of n, a(n) for n = 0..201</a>
%H A347813 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%H A347813 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%e A347813 a(1) = 19:
%e A347813   ((1,1,1), (0,0,0)),
%e A347813   ((1,1,1), (0,0,1), (0,0,0)),
%e A347813   ((1,1,1), (0,1,0), (0,0,0)),
%e A347813   ((1,1,1), (0,1,1), (0,0,0)),
%e A347813   ((1,1,1), (1,0,0), (0,0,0)),
%e A347813   ((1,1,1), (1,0,1), (0,0,0)),
%e A347813   ((1,1,1), (1,1,0), (0,0,0)),
%e A347813   ((1,1,1), (0,1,1), (-1,0,0), (0,0,0)),
%e A347813   ((1,1,1), (0,1,1), (0,0,1), (0,0,0)),
%e A347813   ((1,1,1), (0,1,1), (0,1,0), (0,0,0)),
%e A347813   ((1,1,1), (0,1,1), (1,0,0), (0,0,0)),
%e A347813   ((1,1,1), (1,0,1), (0,-1,0), (0,0,0)),
%e A347813   ((1,1,1), (1,0,1), (0,0,1), (0,0,0)),
%e A347813   ((1,1,1), (1,0,1), (0,1,0), (0,0,0)),
%e A347813   ((1,1,1), (1,0,1), (1,0,0), (0,0,0)),
%e A347813   ((1,1,1), (1,1,0), (0,0,-1), (0,0,0)),
%e A347813   ((1,1,1), (1,1,0), (0,0,1), (0,0,0)),
%e A347813   ((1,1,1), (1,1,0), (0,1,0), (0,0,0)),
%e A347813   ((1,1,1), (1,1,0), (1,0,0), (0,0,0)).
%p A347813 s:= proc(n) option remember;
%p A347813      `if`(n=0, [[]], map(x-> seq([x[], i], i=-1..1), s(n-1)))
%p A347813     end:
%p A347813 b:= proc(l) option remember; (n-> `if`(l=[0$n], 1, add((h-> `if`(
%p A347813       add(i^2, i=h)<add(i^2, i=l), b(sort(h)), 0))(l+x), x=s(n))))(nops(l))
%p A347813     end:
%p A347813 a:= n-> b([n$3]):
%p A347813 seq(a(n), n=0..12);
%t A347813 s[n_] := s[n] = If[n == 0, {{}}, Sequence @@ Table[Append[#, i], {i, -1, 1}]& /@ s[n-1]];
%t A347813 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 A347813 a[n_] := b[{n, n, n}];
%t A347813 Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Nov 04 2021, after _Alois P. Heinz_ *)
%Y A347813 Column k=3 of A347811.
%Y A347813 Cf. A348201.
%K A347813 nonn,walk
%O A347813 0,2
%A A347813 _Alois P. Heinz_, Sep 14 2021