A322419 Number of n-step self-avoiding walks on L-lattice.

1, 2, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356, 564, 904, 1448, 2320, 3684, 5872, 9376, 14960, 23688, 37652, 59912, 95316, 150744, 239080, 379528, 602424, 951788, 1507136, 2388252, 3784344, 5973988, 9447880, 14950796, 23658540, 37321752, 58965260, 93206864, 147333080, 232286272

Offset

0,2

Comments

The L-lattice is an oriented square lattice in which each step must be followed by a step perpendicular to the preceding one.

Links

Sean A. Irvine, Table of n, a(n) for n = 0..50

Robert FERREOL, The a(4)=12 walks in L-lattice

Keh-Ying Lin and Yee-Mou Kao, Universal amplitude combinations for self-avoiding walks and polygons on directed lattices, J. Phys. A: Math. Gen. 32 (1999), page 6929.

A. Malakis, Self-avoiding walks on oriented square lattices, Journal of Physics A: Mathematical and General, Volume 8, Number 12 (1975), page 1890.

Wikipedia, Connective constant

Formula

a(n) = 4*A189722(n) for n >= 2.

It is proved that a(n)^(1/n) has a limit mu called the "connective constant" of the L-lattice; approximate value of mu: 1.5657. It is only conjectured that a(n + 1) ~ mu * a(n).

Example

a(1) = 2 because there are only two possible directions at each intersection; for the same reason a(2) = 2*2 and a(3) = 2*4 ; but a(4) = 12 (not 16) because four paths return to the starting point and are not self-avoiding. See the 12 paths under "links".

Maple

walks:=proc(n)
    option remember;
    local i, father, End, X, walkN, dir, u, x, y;
    if n=1 then [[[0, 0]]] else
         father:=walks(n-1):
         walkN:=NULL:
         for i to nops(father) do
            u:=father[i]:End:=u[n-1]:if n mod 2 = 0 then
            dir:=[[1, 0], [-1, 0]] else dir := [[0, 1], [0, -1]] fi:
            for X in dir do
             if not(member(End+X, u)) then walkN:=walkN, [op(u), End+X] fi;
             od od:
         [walkN] fi end:
n:=5:L:=walks(n):N:=nops(L);
# This program explicitly gives the a(n) walks.

Mathematica

mo = {{1, 0}, {-1, 0}}; moo = {{0, 1}, {0, -1}}; a[0] = 1;
a[tg_, p_: {{0, 0}}] := Module[{e, mv},
If[Mod[tg, 2] == 0, mv = Complement[Last[p] + # & /@ mo, p],
mv = Complement[Last[p] + # & /@ moo, p]];
If[tg == 1, Length@mv, Sum[a[tg - 1, Append[p, e]], {e, mv}]]];
a /@ Range[0, 20] (* after the program from Giovanni Resta at A001411 *)

Prog

(Python)
def add(L, x):
    M = [y for y in L]
    M.append(x)
    return M
plus = lambda L, M: [x + y for x, y in zip(L, M)]
mo = [[1, 0], [-1, 0]]
moo = [[0, 1], [0, -1]]
def a(n, P=[[0, 0]]):
    if n == 0:
        return 1
    if n % 2 == 0:
        mv1 = [plus(P[-1], x) for x in mo]
    else:
        mv1 = [plus(P[-1], x) for x in moo]
    mv2 = [x for x in mv1 if x not in P]
    if n == 1:
        return len(mv2)
    else:
        return sum(a(n - 1, add(P, x)) for x in mv2)
[a(n) for n in range(21)]

Crossrefs

Cf. A001411 (square lattice), A117633 (Manhattan lattice), A189722, A004277 (coordination sequence), A151798.

Sequence in context: A100684 A368430 A131770 * A246850 A294066 A163489

Adjacent sequences: A322416 A322417 A322418 * A322420 A322421 A322422

Keyword

nonn,walk

Author

Robert FERREOL, Dec 07 2018

Status

approved