Lucas Strings

[Creighton Dement]

Dear Seqfans, 

First, I would like to say "Happy New Year" to all and express my
agreement on how much easier it has become to search the OEIS. No longer
would I need to take the day off to see who might have contributed to
one of my favorite submitted sequences. A thought for sometime in the
future (which has probably already been mentioned): it would be really
neat if contributers had their own "OEIS blog-account" where they would
be in allowed to log on and save a little information for others to see
about sequences /problems they are currently working on (I guess that's
currently what home pages are for).  This, of course, would be
completely seperate from the main database. 

On "Lucas Strings": I introduced this term a few months back and am now
getting round to examining it in more detail.  Upon "googling" a half
hour or so ago, I saw the term has already been introduced 
http://www.u-bourgogne.fr/LE2I/jl.baril/minimal.pdf

Though I've only glanced over the first pages of the article (it would
presumably take me a very long time to understand in full),
surprisingly, it seems the "n-th Lucas String" refered to by me is quite
similar to the "order 2 length n Lucas String" referred to there (so,
hopefully, I can keep the terminology - which seems quite appropriate-
without causing too much confusion) . 

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? 

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.