A131709

[kohmoto]

----- Original Message ----- 
From: "Max Alekseyev" <maxale at gmail.com>
To: "koh" <zbi74583 at boat.zero.ad.jp>
Cc: <seqfan at ext.jussieu.fr>
Sent: Saturday, November 03, 2007 7:46 AM
Subject: Re: RE : A131709


> On 10/31/07, koh <zbi74583 at boat.zero.ad.jp> wrote:
>
>> > The total number of partitions (i.e., a(4) in A131709) is
>> > 4 + 4 + 32 + 4 + 61 = 105
>> >
>>     I think that the last term is 60 , so it bedcomes 104.
>
> Agree.
>
>>     My result corresponded with your classification is as follows.
>>     a(3) = 3^4 + 2*(3^2-1) - 1 + 2+ 3 + 3
>>            = {5)+21} + {3)-16} - {I} + {2)-2} + {4)-1} +  {1)-1}
>>     Where,"I" means the " impossible case".
>>     I didn't counted it.
>>
>>     I have awoke that I counted twice the case of  two 4-cycle.
>>     So, the second term must be 2*(3^2-1)  = 16
>>     Hence a(3) = 104
>>
>>     Both results are the same.
>
> Yes, a(3)=104 and I suggest the following recurrent formula:
> a(n+1) = 10*(a(n)-a(n-1)) + 4
> giving
> 4, 14, 104, 904, 8004, ...
>
    How did you get such an easy formula?
    Could you explain about it?

> [...]
>
>>     a(4)
>>     = 3^6 + 2*3^4 - 2*3^2 + 2*3^2 - 1 + 2*1 + 3*3^2 + 2*1 + 2*3 + 3
>>     = 729 + 162 + 1 + 27 + 11
>>     = 930
>>
>>     It is not correct.
>>     I counted the cycles twice in 2nd and 3rd and 4th term and mistook 
>> the sign of 6th term.
>>
>>     The total number is the following.
>>     = 3^6 + 2*(3^4-2) - 2*(3^2-1) + 2*(3^2-1)  - 1 -2*1 + 3*3^2 + 2*1 + 
>> 2*3 + 3
>>     = 922
>>
>>     Corresponding cycles :
>>         nc      14c              14c              16c            16c  24c 
>> 24c         18c     16c+14c   1.10c
>>         where "nc" means No cycle , "14c" means One 4 cyclth.
>
> I counted yet another way and still got a(4)=904 that agrees with the
> recurrence above.
> Please double check your computation of a(4).
>
    I seem to mistake when I count the case of two cycles.
    I found second term shuld be 2(3^4-10).
    So, my result is 906.
    I counted it five times.


    Could any other member of Seqfan tell me a exact number for a(4)?




> Regards,
> Max
>

    By the way, I think that Neil don't read our mail about A131709 at all.
    The description in A131709 is still nonsense and has nothing about "Bus 
route".
    We must send him a mail in which we should write a hundred of claims.

    Yasutoshi
 




Dear Yasutoshi,   You said:

> By the way, I think that Neil don't read our mail about A131709 at all.

Not true!   In fact I asked Max to send me a better version,
and he said, as I recall, that you two had not
yet agreed on the terms of this sequence.
The discussions in the emails you posted were not clear to me.

Let me repeat:  the best way to make corrections is to
use the web page for sending in comments.  It is certainly risky
to assume that I read every email that is posted to
the seqfan mailing list!

Best regards

Neil