0,2

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

A prudent walk never takes a step pointing towards a vertex it has already visited. Prudent walks are self-avoiding but not reversible in general.

Alois P. Heinz, Table of n, a(n) for n = 0..61

Mireille Bousquet-Mélou, Families of prudent self-avoiding walks, arXiv:0804.4843. J. Combin. Theory Ser. A 117 no. 3 (2010) 313-344.

Enrica Duchi, On some classes of prudent walks, in: FPSAC'05, Taormina, Italy, 2005.

Wikipedia, Self-avoiding walk

Two 5-step self-avoiding walks on hexagonal lattice from (S) to (E), the walk at left is prudent while the walk at right is not prudent:

. o---o...o...o...o---o

. . \ . \ . . . . . \ . \

. o..(S)..o...o...o..(E)..o

. . . . / . . . . . . . /

. (E)--o...o...o..(S)--o

i:= n-> max(n, 0)+1: d:= n-> max(n-1, -1):

b:= proc(n, x, y, z, u, v, w) option remember;

`if`(n=0, 1, `if`(x>y, b(n, y, x, w, v, u, z),

b(n-1, d(x), d(y), z, i(u), i(v), w)+

`if`(min(y, z)<=0 or x=-1,

b(n-1, d(y), d(z), u, i(v), i(w), x), 0)+

`if`(min(z, u)<=0 or y=-1,

b(n-1, d(z), d(u), v, i(w), i(x), y), 0)+

`if`(min(v, w)<=0 or x=-1,

b(n-1, d(v), d(w), x, i(y), i(z), u), 0)+

`if`(min(w, x)<=0 or y=-1,

b(n-1, d(w), d(x), y, i(z), i(u), v), 0)))

end:

a:= n-> `if`(n=0, 1, 6*b(n-1, -1$2, 0, 1$2, 0)):

seq(a(n), n=0..20);

i[n_]:= Max[n, 0]+1; d[n_]:= Max[n-1, -1];

b[n_, x_, y_, z_, u_, v_, w_] := b[n, x, y, z, u, v, w] = If[n==0, 1, If[x>y, b[n, y, x, w, v, u, z], b[n-1, d[x], d[y], z, i[u], i[v], w]+ If[Min[y, z]<=0 || x==-1, b[n-1, d[y], d[z], u, i[v], i[w], x], 0]+ If[Min[z, u]<=0 || y==-1, b[n-1, d[z], d[u], v, i[w], i[x], y], 0]+ If[Min[v, w]<=0 || x==-1, b[n-1, d[v], d[w], x, i[y], i[z], u], 0]+ If[Min[w, x]<=0 || y==-1, b[n-1, d[w], d[x], y, i[z], i[u], v], 0]]];

a[n_]:= If[n==0, 1, 6*b[n-1, -1, -1, 0, 1, 1, 0]];

Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 10 2017, translated from Maple *)

Cf. A001334, A192871.

Sequence in context: A034545 A002920 A255463 * A001334 A125316 A092439

Adjacent sequences: A192205 A192206 A192207 * A192209 A192210 A192211

nonn,walk

Alois P. Heinz, Jul 05 2011

approved