Lucas Strings

[Creighton Dement]

- - - - - - - - - - - - - - - - - - - - - - - - - - - - -

> Date: Sat, 31 Dec 2005 13:28:59 +0100
> Subject: Lucas Strings
> From: "Creighton Dement" <crowdog at crowdog.de>
> To: seqfan at ext.jussieu.fr

> Dear Seqfans,
> 
[snip]
>
> The new paper "Floretion-generated Integer Sequences"
> http://crowdog.de/Flointseq.pdf (which should probably be relabled
> "Floretion-generated 2nd Order Recurrence Sequences" as there is
> already almost too much to talk about for one paper-  without going
> into the 4th and higher order recurrence relations) has been updated
> to cover this topic
> 
> 
> Using the definitions given in the chapter "Necklaces" of the above
> paper ( I just wrote everything down over the last two nights so I
> hope anyone will please tell me if I've made mistakes and / or left
> out important information)  I counted the total number of unmatched
> b's in the n-th Lucas String (beginning with the 3rd), and got the
> sequence  4, 6, 9, 14, 19, 30, 44, 68, 99, 168, 245, 402, 636, 1026,
> 1613, 2650 [Dirichlet convolution of Fibonacci numbers with phi(n),
> A034748] 
> Next, I counted the total number of ab's in the n-th Lucas String
> (beginning with the 3rd):  1, 3, 3, 8, 8, 17, 23, 41, 55, 102, 144,
> 247, 387, 631, 987, 1636
> This sequence is unlisted (Superseeker: no ideas). Can someone give an
> alternative  description of this sequence?
> 

I forgot to mention that the first time I saw the sequence, I noticed it
contained several Fibonacci numbers. I looked at it closer, but couldn't
see an immediate pattern and quickly gave up. After I sent the last
post,  I got a reply from Antti suggesting that I start by analysing the
fact that it contains Fibonacci numbers.  After a bit of thought, I
quickly came to the conclusion: "If Antti said it... better look at it
again".

The sequence is
(beginning with the 3rd):  1, 3, 3, 8, 8, 17, 23, 41, 55, 102, 144,
 247, 387, 631, 987, 1636

Since I began with the 3rd term, set a(3) = 1, a(4) = 3, .... 

Then it indeed appears that a "pattern" is present:

Fib(3-1) = 1 = a(3) 
Fib(5-1) = 3 = a(5) 
Fib(7-1) = 8 = a(7) 
Fib(11-1) = 55 = a(11)
Fib(13-1) = 144 = a(13) 
Fib(17-1) =  987 = a(17) 

Conjecture: for prime p > 2 : Fib(p-1) = a(p)
 
Note: I defined the sequence (a(n)) as "total number of ab's in the n-th
Lucas string" and  stated that  "the total number of unmatched
b's in the n-th Lucas String" (beginning with the 3rd) is the sequence 
4, 6, 9, 14, 19, 30, 44, 68, 99, 168, 245, 402, 636, 1026,  

The total number of ab's in the n-th Lucas string is by definition the
same as "total number of matched b's in the n-th Lucas string.".


> 1613, 2650 [Dirichlet convolution of Fibonacci numbers with phi(n),
> A034748] 
> Here is an entire list of the Lucas Strings up to n = 16.
> http://www.crowdog.de/LucasStrings.html
> There appear to be several integer sequences of interest "tucked away"
> on that page.
> 
> Many thanks,
> Creighton
> 
> -It's a shame when the girl of your dreams would still rather be with
> someone else when you're actually in a dream.
> 
> 
> 
>