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

Eric Angelini Eric.Angelini at kntv.be
Thu Oct 2 18:13:43 CEST 2014

```Hello Aai,
I've had the too the idea of "rounding" -- but discarded it
because it is difficult to be explained in detail -- and
for every case.
Best,
É.

-----Message d'origine-----
De : SeqFan [mailto:seqfan-bounces at list.seqfan.eu] De la part de Aai
Envoyé : jeudi 2 octobre 2014 18:04
À : Sequence Fanatics Discussion list
Objet : [seqfan] Re: Brute force density: triples and cubes

Also interesting:

- use 50 51 52 53 54 as starting values
- with each value generate a sequence, group on trailing 0,
take the sum of each group and divide by 27 (max multiplication
result) (*)

(*) some starting values have to be round up to one.

Here are the resulting sequences:

50: 1 1 1 2 3 4 6 9 13 19 28 41 60 88 129 189 277 406 595 872

similar to ==> A000930 Narayana's cows sequence

51: 1 1 2 3 4 6 9 13 19 28 41 60 88 129 189 277 406 595 872 1278 shifted previous

52: 1 2 2 3 5 7 10 15 22 32 47 69 101 148 217 318 466 683 1001 1467

ignore first 4 ==>
A020711     Pisot sequences E(5,7), P(5,7)

53: 1 2 2 3 5 7 10 15 22 32 47 69 101 148 217 318 466 683 1001 1467 equals the previousone

54: 1 1 2 4 5 7 11 16 23 34 50 73 107 157 230 337 494 724 1061 1555

ignore first 4 ==> looks like
A164316 Number of binary strings of length n with no substrings equal to
000 001 or 010

> 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

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