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

Aai agroeneveld400 at gmail.com
Thu Oct 2 18:32:32 CEST 2014

```Correction.

group on trailing 0 only counts for starting value 50.

In fact I don't group on a particular value. The grouping is part of the
iterating process I use, e.g for 50 the iteration goes as follows:

50
15 0
3 15 0
9 3 15 0
27 9 3 15 0

for 53 I get:

53
15 9
3 15 27
9 3 15 6 21
27 9 3 15 18 6 3

Sorry for the noise.

On 02-10-14 18:03, Aai wrote:
> 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.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

```