A328269 - OEIS

%I #26 Oct 15 2020 12:37:33

%S 1,3,26,343,5594,103730,2094028,44889351,1006126370,23337166962,

%T 556199376622,13550764116530,336190200180652,8468872074477060,

%U 216120719672921820,5577150906683145103,145324963753397617230,3819107708757101038562,101122686499165125017886

%N Number of walks on cubic lattice starting at (0,0,0), ending at (0,n,n), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).

%H Alois P. Heinz, <a href="/A328269/b328269.txt">Table of n, a(n) for n = 0..639</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>

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

%F a(n) = A328300(2n,n).

%F a(n) is odd <=> n in { A000225 }.

%F a(n) ~ c * 2^(3*n) * (2 + sqrt(3))^n / n^2, where c =

%F 0.081957778985952080274457799679795068000445171394180053136120884510526907545... - _Vaclav Kotesovec_, May 10 2020

%e a(1) = 3: [(0,0,0),(1,0,0),(0,1,1)], [(0,0,0),(0,1,0),(0,1,1)], [(0,0,0),(0,0,1),(0,1,1)].

%e a(2) = 26: [(0,0,0),(1,0,0),(2,0,0),(1,1,1),(0,2,2)], [(0,0,0),(1,0,0),(1,1,0),(1,1,1),(0,2,2)], ..., [(0,0,0),(0,0,1),(0,1,1),(0,1,2),(0,2,2)], [(0,0,0),(0,0,1),(0,0,2),(0,1,2),(0,2,2)].

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

%p       add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(

%p       sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))

%p     end:

%p a:= n-> b([0, n$2]):

%p seq(a(n), n=0..23);

%t b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];

%t a[n_] := b[{0, n, n}];

%t a /@ Range[0, 23] (* _Jean-François Alcover_, May 12 2020, after Maple *)

%Y Bisection (even part) of A328280.

%Y Cf. A000225, A000984, A277262, A328270, A328300.

%K nonn,walk

%O 0,2

%A _Alois P. Heinz_, Oct 10 2019