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A006851 Trails of length n on honeycomb lattice.
(Formerly M2560)
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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:1007.0575v2)

A. J. Guttmann, Lattice trails II: numerical results, J. Phys. A 18 (1985), 575-588.

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-Fran├žois Alcover, Mar 20 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A001668.

Sequence in context: A090572 A163876 A033893 * A164352 A115829 A115805

Adjacent sequences:  A006848 A006849 A006850 * A006852 A006853 A006854

KEYWORD

nonn,walk

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 22 19:11 EDT 2021. Contains 343177 sequences. (Running on oeis4.)