A006851 Trails of length n on honeycomb lattice.

1, 3, 6, 12, 24, 48, 96, 186, 360, 696, 1344, 2562, 4872, 9288, 17664, 33384, 63120, 119280, 225072, 423630, 797400, 1499256, 2817216, 5286480, 9918768, 18592080, 34840848, 65228874, 122105496, 228402168, 427176336, 798373662, 1491985800, 2786515176, 5203816992, 9712725234, 18127267800

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..43
• H. Duminil-Copin and S. Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt(2)), arXiv:1007.0575 [math-ph], 2010-2011.
• A. J. Guttmann, Lattice trails II: numerical results, J. Phys. A 18 (1985), 575-588.

Crossrefs

Cf. A001668.

Programs

Maple

a:= proc(n) option remember; local v, b;
      if n<2 then return 1 +2*n fi;
      v:= proc() false end: v(1, 0):= true;
      b:= proc(n, d, x, y) local c;
            if v(x, y) then `if`(n>0 or [x, y, d]=[1, 0, 1], 0, 1)
          elif n=0 then 1
          else v(x, y):= true;
               c:= b(n-1, [$2..6, 1][d], x+[0, -1, -1, 0, 1, 1][d],
                                         y+[1, 1, 0, -1, -1, 0][d])+
                   b(n-1, [6, $1..5][d], x+[1, 1, 0, -1, -1, 0][d],
                                         y+[-1, 0, 1, 1, 0, -1][d]);
               v(x, y):= false; c
            fi
          end;
      6*b(n-2, 2, 1, 1)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 08 2011

Mathematica

a[n_] := a[n] = Module[{v, b}, If[n<2, Return[1+2*n]]; v[, ] = False; v[1, 0] = True; b[n0_, d_, x_, y_] := Module[{c}, Which[v[x, y], If[n0>0 || {x, y, d} == {1, 0, 1}, 0, 1], n0 == 0, 1, True, v[x, y] = True; c = b[n0-1, {2, 3, 4, 5, 6, 1}[[d]], x+{0, -1, -1, 0, 1, 1}[[d]], y+{1, 1, 0, -1, -1, 0}[[d]]] + b[n0-1, {6, 1, 2, 3, 4, 5}[[d]], x+{1, 1, 0, -1, -1, 0}[[d]], y+{-1, 0, 1, 1, 0, -1}[[d]]]; v[x, y] = False; c]]; 6*b[n-2, 2, 1, 1]]; Table[Print[a[n]]; a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)