# [seqfan] Re: confused about toothpick sequence A139250!

Benoît Jubin benoit.jubin at gmail.com
Tue Apr 14 06:07:04 CEST 2009

```There is something curious about this toothpick sequence A139250.  Its
sequence of first differences, A139251, is of course divisible by
four, giving A152978.  It is easy to prove (with a good offset) that
A152978(2^n+k) does not depend on n if k<=2^(n-1), and the latter is
given by A152980, which has a similar property, with "limit sequence"
A153001, which again has a similar property (but this time, the limit
sequence doesn't seem to be in the OEIS).  This hints at a formula for
these sequences where the expression of n in base 2 plays a role
(although it seems that the "limit sequence" appears further and
further).
I'm sorry but I have no time to check this further.  Maybe someone
will find something interesting in this direction.

Benoit

On Mon, Apr 13, 2009 at 5:42 PM, Maximilian Hasler
<maximilian.hasler at gmail.com> wrote:
>
> On Mon, Apr 13, 2009 at 5:37 PM, N. J. A. Sloane <njas at research.att.com> wrote:
> > I was thinking the new toothpicks would have to touch the midpoints
> > of old toothpicks, not their endpoints (otherwise we could add 4
> > at the first step)
>
> I think this is correct.
> For me the values are reproducible. Below a PARI program which prints
> the following:
>
> 0,1,3,7,11,15,23,35,43,47,55,67,79,95,123,155,171,175,183,195,207,223,251,283,303,
> 319,347,383,423,483,571,651,683,687,695,707,719,735,763,795,815,831,859,895,935,
> 995,1083,1163,1199,1215,1243,1279,1319,1379,1467,1551,1607,1667,1759,1871,
> 2011,229,2475,2667,2731,2735,2743,2755,2767,2783,2811,2843,2863,2879,2907,
> 2943,2983,3043,3131,3211,3247,3263,3291,3327,3367,3427,3515,3599,3655,3715,
> 3807,3919,4059,4267,4523,4715,4783,4799,4827,4863,4903,
>
> p=Set(0); /* The set of "used" points. Points are written as complex
> numbers, c=x+iy.
> To avoid fractions, I take the toothpicks of length 2. */
> ne=[[1,1],[-1,-1]]; /* the [new] list of (exposed) endpoints. Exposed
> endpoints are listed as [c,d] where c=x+iy is the position of the
> endpoint, and d=a+ib is the direction. */
> print1("0,"cnt=1);
> {for(n=1,99,
>  ee=eval(Set(ne)); /* set of exposed endpoints [use set for sorting] */
>  /* remove those which actually are not exposed */
>  forstep( k=#ee, 2, -1, ee[k][1]==ee[k-1][1] | next;
> ee=vecextract(ee,Str("^",k-1,".."k)); k--);
>  print1(","cnt+=#ee);  /* for each exposed endpoint, a toothpick will
> be added in this move */
>  ne=Set(Mat(ee~)[,1]); /* discard directions */
>  p=setunion(p, ne);  /* add endpoints from last move to "used" points */
>  ne=[]; /* new (exposed) endpoints */
>  for( k=1, #ee,  /* add exposed endpoints of new toothpicks */ ep=ee[k];
>    !setsearch(p, ep[1]+I*ep[2]) & ne=concat(ne,[[ep[1]+I*ep[2],I*ep[2]]]);
>    !setsearch(p, ep[1]-I*ep[2]) & ne=concat(ne, [[ep[1]-I*ep[2],-I*ep[2]]]);
> ) )}
>
> Maximilian
>
>
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```