3D random walks

[Wouter Meeussen]

At 22:52 6-06-98 +0200, Wouter Meeussen wrote:
>hi all,
> ... clip
>Random walks that self-intercept form a closed loop at that instant.
>For which integers does the walk self-intercept?


***  yes, they self-intercept somewhere, but they do not form closed loops!
***  they *end* in a loop, but can have a "irrelevant" walk before entering it.


>Starting from coordinates {0,0,0}, we fix the direction of the first step as
>being {1,1,-1). From there on, the choice between subsequent steps {5,6,7}
>will be determined by the ternary representation of our input number.
> ... clip
>Using this scheme, we find closed loops for n=
>
> 
>{286,312,390,443,468,521,599,625,677,703,755,781,782,807,833,858,859,885,936,
>  937,938,963,1015,1016,1041,1119,1170,1171,1172,1197,1249,1250,1328,1329,
> ... clip
>  3801,3853,3879,3896,3904,3905,3931,3957,3960,3983,3984,3985,3986,3987,3988,
>  3989, ... }
>
>Question :
>        Is this a bona-fide "informative" integer sequence?
>        Any suggestions? 
>
>wouter.
>
>
>Dr. Wouter L. J. MEEUSSEN
>w.meeussen.vdmcc at vandemoortele.be
>eu000949 at pophost.eunet.be
>
>
>

you might want to reconsider the issue.


How about the following ones for *real* closed loops?
they can be recognised by the fact that they intersect exactly in {0,0,0},
the starting point (and nowhere else) :

five steps (hexagons): (3 of them)
{104,130,156}
six steps (heptagons): (6 of them)
{960,1025,1200,1240,1320,1435}
seven steps (octagons):
{}
eight steps (nonagons): (65 of them)
{6776,6800,6992,7018,7260,7502,7744,7770,7962,7986,8228,8306,8470,8496,8548,
  8712,8723,8954,8978,9124,9196,9366,9438,9455,9680,9922,10065,10156,10164,
  10406,10431,10484,10648,10726,10882,10890,11132,11285,11374,11407,11616,
  11640,11858,11930,12100,12108,12172,12342,12584,12610,12749,12826,12834,
  12993,13068,13310,13336,13552,13603,13794,13818,14036,14091,14278,14520}
nine steps (decagons):
{}
ten steps (undecagons): (307 of them)
Out[158]=
{60713,60737,60883,60955,61224,61721,61934,62395,62446,62662,62688,62899,
  63014,63094,63134,63665,63907,64581,64605,65085,65102,65174,65253,65318,
  65589,67271,67542,67607,67686,67758,67775,68255,68279,68953,69195,69726,
  69766,69846,69961,70172,70198,70414,70465,70926,71139,71636,71905,71977,
  72123,72147,73999,74025,74132,74241,74252,74292,74492,74725,74837,74967,
  76203,76318,76398,76438,76678,77023,77172,77406,77656,77708,77734,77898,
  77976,78132,78293,78382,78535,78624,78657,79217,79892,79918,80557,80669,
  80826,80885,81323,81395,81608,82048,82088,82613,82688,82847,82855,82932,
  83071,83097,83339,83509,83573,83581,84207,84272,84543,84848,85014,85025,
  85191,85751,86721,87144,87203,87281,87360,87377,87857,87929,87953,89137,
  89215,89247,89536,89544,89863,89896,90016,90345,90616,90744,91315,91323,
  91651,91722,92041,92523,93563,93627,93908,93972,94148,94268,95376,95749,
  95805,95861,96078,96094,96102,96334,96454,96587,96828,96903,97062,97148,
  97174,97313,97554,97788,97967,98039,98209,98256,98280,98656,99332,99358,
  100100,100108,100264,100342,100506,100559,100584,100826,100834,100925,
  101068,101536,101651,102019,102286,102419,102612,102745,103012,103111,
  103740,103812,103953,104463,104522,104679,104791,105161,105216,105918,
  106641,106652,106816,106868,106894,107058,107136,107394,107459,107871,
  108096,108113,108185,108811,108819,109147,109218,109578,109618,109795,
  110274,110298,110346,110997,111109,111165,111325,111333,111964,112452,
  113284,113324,113404,113519,114476,114671,114979,115204,115463,115535,
  115681,115705,115776,115936,116111,116136,116189,116353,116431,116664,
  116856,116915,117157,117641,117667,117699,117859,117883,117891,119761,
  119876,119956,119996,120527,120769,120956,121011,121229,121253,121684,
  121940,122093,122182,122215,122457,122713,123414,123627,124124,124196,
  124393,124611,124635,126348,126413,126732,126797,127061,127085,127277,
  127303,127545,127787,128532,128572,128749,128892,128983,128991,129220,
  129271,129948,129972,130711,130839,130927,130935,130999,131169}

with sympathy,

wouter.

*******************************************************************
 
walk[vec_]:=Module[{},
                la={1,1,-1}   ;(* wa= the walk, la=the last step *) 
                wa={{0,0,0},la};len=Length[vec];old=1;bump=False;
		i=1;
		While[!bump &&i<= len ,
                   la=la+steps[[new=idx[[old,vec[[i]]   ]] ]];
                   If[!FreeQ[wa,la],bump=True];
		   wa={wa,la };
		   old=new;
                i++];
             (* Partition[Flatten[wa],3] *) ;
             la . la ]

little program :

   out={};
   Do[
     If[walk[Reverse[IntegerDigits[k,3]+1]]===0&& i-1=== len,AppendTo[out,k]],
   {k,3^9, 3^10 -1 }];
   out
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be