[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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