# [seqfan] Re: Need proof that Kimberling's A26185 = A198173

Neil Sloane njasloane at gmail.com
Wed Feb 5 21:42:42 CET 2020

```Kevin, I took care of that!  My notes:  6 more Kimberling dupes
26177=26196=26216,  26217=26178=26197.

On Mon, Feb 3, 2020 at 10:23 PM Neil Sloane <njasloane at gmail.com> wrote:

> Kevin, great, thanks!  I thought there were probably more of these
> duplicates but I didn't have time to do a search.
> I'll do the merging over the next couple of days
>
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Email: njasloane at gmail.com
>
>
>
> On Mon, Feb 3, 2020 at 7:54 PM Kevin Ryde via SeqFan <
> seqfan at list.seqfan.eu> wrote:
>
>> njasloane at gmail.com (Neil Sloane) writes:
>> >
>> > I have now processed all these pairs or triples or even quadruples ...
>>
>> Also maybe A026177 = A026196 = A026216 ?  A026177 is Michel Dekking's
>> Pi_odd in section 2.3.  I had some work-in-progress bits for that
>> one, which could be in that section already.  I hadn't thought of
>> distinguishing type III and IV but instead went ceil(2n/3).
>>
>>     A026177
>>     Let d=A060236(n) be the lowest non-0 ternary digit of n.
>>       If d=1 then a(n)=ceil(2n/3), otherwise a(n)=2n.
>>     a(n) = ceil(2n / 3^A137893(n)).
>>     a(3n + (1 if n=1 mod 3)) = 3*a(n) is all multiples of 3.
>>     a(3n) = 3*a(n) - (1 if n=1 mod 3).
>>
>> The second last is "sequence is its own multiples of 3" giving
>> A026177 = A026216, after suitable explanation.  I don't know how to
>> get the further apparent A026177 = A026196; the latter coming from
>> Pi_42 of section 2.4.
>>
>> The inverses of each would be A026178 = A026197 = A026217 which
>> R. J. Mathar already contemplates in A026217.
>>
>> (I arrived at A026177 since its choice of 2n/3 or 2n goes according
>> to terdragon left or right turn.  New high values and positions, and
>> increments between those, are then turn or segment expansion related.)
>>
>> --
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>>
>

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