A322419 - OEIS

%I #49 Jul 29 2020 23:06:11

%S 1,2,4,8,12,20,32,52,84,136,220,356,564,904,1448,2320,3684,5872,9376,

%T 14960,23688,37652,59912,95316,150744,239080,379528,602424,951788,

%U 1507136,2388252,3784344,5973988,9447880,14950796,23658540,37321752,58965260,93206864,147333080,232286272

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

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

%H Sean A. Irvine, <a href="/A322419/b322419.txt">Table of n, a(n) for n = 0..50</a>

%H Robert FERREOL, <a href="/A322419/a322419_1.gif">The a(4)=12 walks in L-lattice</a>

%H Keh-Ying Lin and Yee-Mou Kao, <a href="https://doi.org/10.1088/0305-4470/32/40/303">Universal amplitude combinations for self-avoiding walks and polygons on directed lattices</a>, J. Phys. A: Math. Gen. 32 (1999), page 6929.

%H A. Malakis, <a href="http://iopscience.iop.org/article/10.1088/0305-4470/8/12/007/meta">Self-avoiding walks on oriented square lattices</a>, Journal of Physics A: Mathematical and General, Volume 8, Number 12 (1975), page 1890.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Connective_constant">Connective constant</a>

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

%F 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).

%e 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".

%p walks:=proc(n)

%p     option remember;

%p     local i,father,End,X,walkN,dir,u,x,y;

%p     if n=1 then [[[0,0]]] else

%p          father:=walks(n-1):

%p          walkN:=NULL:

%p          for i to nops(father) do

%p             u:=father[i]:End:=u[n-1]:if n mod 2 = 0 then

%p             dir:=[[1,0], [-1, 0]] else dir := [[0,1], [0, -1]] fi:

%p             for X in dir do

%p              if not(member(End+X,u)) then walkN:=walkN,[op(u),End+X] fi;

%p              od od:

%p          [walkN] fi end:

%p n:=5:L:=walks(n):N:=nops(L);

%p # This program explicitly gives the a(n) walks.

%t mo = {{1, 0}, {-1, 0}}; moo = {{0, 1}, {0, -1}}; a[0] = 1;

%t a[tg_, p_: {{0, 0}}] := Module[{e, mv},

%t If[Mod[tg, 2] == 0, mv = Complement[Last[p] + # & /@ mo, p],

%t mv = Complement[Last[p] + # & /@ moo, p]];

%t If[tg == 1, Length@mv, Sum[a[tg - 1, Append[p, e]], {e, mv}]]];

%t a /@ Range[0, 20] (* after the program from _Giovanni Resta_ at A001411 *)

%o (Python)

%o def add(L, x):

%o     M = [y for y in L]

%o     M.append(x)

%o     return M

%o plus = lambda L, M: [x + y for x, y in zip(L, M)]

%o mo = [[1, 0], [-1, 0]]

%o moo = [[0, 1], [0, -1]]

%o def a(n, P=[[0, 0]]):

%o     if n == 0:

%o         return 1

%o     if n % 2 == 0:

%o         mv1 = [plus(P[-1], x) for x in mo]

%o     else:

%o         mv1 = [plus(P[-1], x) for x in moo]

%o     mv2 = [x for x in mv1 if x not in P]

%o     if n == 1:

%o         return len(mv2)

%o     else:

%o         return sum(a(n - 1, add(P, x)) for x in mv2)

%o [a(n) for n in range(21)]

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

%K nonn,walk

%O 0,2

%A _Robert FERREOL_, Dec 07 2018