[seqfan] Re: Brute force density: triples and cubes

[Aai]

Perhaps a nice link to another sequence.

The first differences of the position of 0's develop as follows:

3 4 5 7 10 14 20 29 42 61 89 130 190 278 407 ...

Except from the first two it looks like:
A020711 Pisot sequences E(5,7), P(5,7)



On 02-10-14 06:53, M. F. Hasler wrote:
> Eric,
> the digit's frequency is not uniform at all.
> I confirm Bob's observation about something like a p-adic convergence
> (i.e., from right to left) of the subsequences between 0's.
> I think it's easy to show that the distance between zeros is strictly
> increasing,
> which excludes existence of a loop,
> and also implies that the asymptotic density of "0" (as well of "5"
> which only occurs immediately before a "0") is zero.
> The other digits seem to occur with relative densities of about
> 19%, 21%, 13%, 7%, 14%, 6%, 10% and 9%.
> This can be computed to arbitrary precision by constructing the
> limiting sequence Bob pointed out,
> ...,3,24,18,9,18,6,3,6,21,27,9,3,15,0.
>
> Below are the first ~900 terms I get (structured to see the limiting
> sequence ending in "0")
> and some PARI code for my records.
>
> Best,
> Maximilian
>
> (PARI)
> EA(n,s=50,d=[])={for(i=1,n,print1(s",");d=concat(d,if(s,digits(s)));s=3*d[1];d=vecextract(d,"^1"));s}
> digit_count(s,c=vector(57))={for(i=1,#s=Vecsmall(s),c[s[i]]++);vecextract(c,"44..")}
>
> 50,
> 15,0,
> 3,[ 15,0,]
> 9,[ 3,15,0,]
> 27,[ 9,3,15,0,]
> 6,21,[ 27,9,3,15,0,]
> 18,6,3,[ 6,21, 27,9,3,15,0,]
> 3,24,18,9,[ 18,6,3,6,21,27,9,3,15,0,]
> 9,6,12,3,24,27,[ 3,24,18,9,18,6,3,6,21,27,9,3,15,0,]
>
> 27,18,3,6,9,6,12,6,21,[
>   9,6,12,3,24,27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]
>
> 6,21,3,24,9,18, 27,18,3,6,18,6,3,[ 27,18,3,6,9,6,12,6,21,
>   9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]
>
> 18,6,3,9,6,12, 27,3,24,6,21,3,24,9,18,3,24,18,9,[
>   6,21,3,24,9,18,27,18,3,6,18,6,3, 27,18,3,6,9,6,12,6,21,
>   9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]
>
> 3,24,18,9,27,18,3,6,6,21,9,6,12,18,6,3,9,6,12,27,3,24,9,6,12,3,24,27,[
>   18,6,3,9,6,12, 27,3,24,6,21,3,24,9,18,3,24,18,9,
>   6,21,3,24,9,18,27,18,3,6,18,6,3, 27,18,3,6,9,6,12,6,21,
>   9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]
>
> 9,6,12,3,24,27,6,21,3,24,9,18,18,6,3,27,18,3,6,3,24,18,9,27,18,3,6,6,21,9,6,12,27,18,3,6,9,6,12,6,21,[
>   3,24,18,9,27,18,3,6,6,21,9,6,12,18,6,3,9,6,12,27,3,24,9,6,12,3,24,27,
>   18,6,3,9,6,12, 27,3,24,6,21,3,24,9,18,3,24,18,9,
>   6,21,3,24,9,18, 27,18,3,6,18,6,3, 27,18,3,6,9,6,12,6,21,
>   9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]
>
> 27,18,3,6,9,6,12,6,21,18,6,3,9,6,12,
> 27,3,24,3,24,18,9,6,21,3,24,9,18,9,6,12,3,24,
> 27,6,21,3,24,9,18,18,6,3, 27,18,3,6,6,21,3,24,9,18, 27,18,3,6,18,6,3,[
>   9,6,12,3,24,27,
> 6,21,3,24,9,18,18,6,3,27,18,3,6,3,24,18,9,27,18,3,6,6,21,9,6,12,27,18,3,6,9,6,12,6,21,
>   3,24,18,9,27,18,3,6,6,21,9,6,12,18,6,3,9,6,12,27,3,24,9,6,12,3,24,27,
>   18,6,3,9,6,12, 27,3,24,6,21,3,24,9,18,3,24,18,9,
>   6,21,3,24,9,18, 27,18,3,6,18,6,3,27,18,3,6,9,6,12,6,21,
>   9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]
>
> 6,21,3,24,9,18, 27,18,3,6,18,6,3,3,24,18,9,
> 27,18,3,6,6,21,9,6,12,9,6,12,3,24, 27,18,6,3,9,6,12, 27,3,24,
> 27,18,3,6,9,6,12,6,21,18,6,3,9,6,12,
> 27,3,24,3,24,18,9,6,21,3,24,9,18,18,6,3,9,6,12,
> 27,3,24,6,21,3,24,9,18,3,24,18,9,[
>   27,18,3,6,9,6,12,6,21,18,6,3,9,6,12, ..., 27,9,3,15,0,]
>
>
> On Wed, Oct 1, 2014 at 9:12 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>> Hello SeqFans,
>> is T containing 10% of 0s, 10% of 1s, 10% of 2s,... 10% of 9s?
>>
>> T starts with 50 and is always extended with the triple of the
>> leftmost digit not yet tripled in T.
>>
>> T = 50,15,0,3,15,0,9,3,15,0,27,9,3,15,0,6,21,27,9,3,15,0,18,6,3,6,21,27,9,3,15,0,3,24,...
>> (if T enters into a loop, the question is closed -- but I can't
>> find one).
>>
>> I would have the same question with Q: Q starts with 90 and is
>> always extended with the cube of the leftmost digit not yet cubed
>> in Q.
>>
>> Q = 90,729,0,343,8,729,0,27,64,27,512,343,8,729,0,8,343,216,64,8,343,125,1,8,27,64,27,...
>>
>> Best,
>> É.
>>
>>
>>
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>
>

-- 
Met vriendelijke groet,
@@i = Arie Groeneveld