[seqfan] Re: Question about submission guidelines

[Brad Klee]

Hi Vladimir,

Your approach also seems good, but requires three recurrence definitions as opposed to two.

If you want to involve Fibonacci, it may be worthwhile to look into the irrational quantity:

n x^n

Where x is the golden ratio.

Best,

Brad

> On Aug 20, 2015, at 3:49 AM, Vladimir Shevelev <shevelev at bgu.ac.il> wrote:
> 
> Hi Bob and Brad,
> 
> Here is quite simple proof.
> Note that F_1=F(n+1)+F(n-1)=L(n), n>=1.
> Indeed, L(1)=F_1(1)=1, L(2)=F_1(2)=3,
> and trivially F_1(n-1)+F_1(n-2)=F_1(n).
> 
> So  F(n)=L(n-1)-F(n-2). Let us add to this an equality
> nF(n)=nF(n-1)+nF(n-2). So, 
> a(n)=a(n-1)+a(n-2)+L(n-1).
> 
> Best regards,
> Vladimir
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Brad Klee [bradklee at gmail.com]
> Sent: 20 August 2015 01:02
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Question about submission guidelines
> 
> Hi Bob,
> 
> I found your relation interesting to prove. It requires some manipulation
> of the connection between recurrence equations. Using "a" for A099920 and
> "L" for A000032 you give
> 
> a(n) = a(n-1) + a(n-2) + L(n-1),          (1)
> 
> or
> 
> L(n-1) = a(n) - a(n-1) - a(n-2).            (2)
> 
> This is a functional definition for L(n) which may or may not be true. By
> the recurrence definition for a(n),
> 
> L(n-1) = a(n-1) - 2 a(n-3) - a(n-4),
> L(n) = a(n) - 2 a(n-2) - a(n-3).               (3)
> 
> These functions match on a few terms, but do they match on all terms?
> Since the Axiom is fine, we just need to check satisfaction of the
> recurrence relations. Combining 1 and 3 by the recurrence definition for
> L(n)
> 
> L(n+1) ?= L(n) +L(n-1),
> L(n+1) ?= 2 a(n) - a(n-1) - 3 a(n-2) - a(n-3).  (4)
> 
> By incrementing Eq. 3
> 
> L(n+1) = a(n+1) - 2 a(n-1) - a(n-2).  (5)
> 
> Subtracting Eq. 5,4
> 
> 0 ?= a(n+1) - 2 a(n) - a(n-1) + 2 a(n-2) + a(n-3).
> 
> According to the recurrence definition of a(n), the equality is satisfied.
> Then the definition of L(n) is proven true.
> 
> This logic affords a production system. We can iterate the recurrence
> relations for L(n) to produce an infinite number of equalities that relate
> L(n+k) to the four values { a(n), a(n-1), a(n-2), a(n-3) }.
> 
> Eq.3 is a nice complement to Eq. 2. Maybe that equation should also be
> added.
> 
> Cheers,
> 
> Brad
> 
> 
> 
> On Wed, Aug 19, 2015 at 12:43 PM, Alonso Del Arte <alonso.delarte at gmail.com>
> wrote:
> 
>> My two cents: if it's important enough for you to write a letter of more
>> than a hundred words to SeqFan about it, it's important enough to submit
>> even now.
>> 
>> Al
>> 
>>> On Wed, Aug 19, 2015 at 1:01 PM, Neil Sloane <njasloane at gmail.com> wrote:
>>> 
>>> sure, post it!
>>> 
>>> 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.
>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>> Email: njasloane at gmail.com
>>> 
>>> 
>>> On Wed, Aug 19, 2015 at 10:47 AM, Bob Selcoe <rselcoe at entouchonline.net>
>>> wrote:
>>> 
>>>> (Hi Olivier - no worries if you would rather reply to me individually
>>> than
>>>> post this to the general discussion board)
>>>> 
>>>> Hello Seqfans,
>>>> 
>>>> I was hoping some of the more seasoned contributors might be able to
>>>> provide a little guidance on what (and what not) to post, given the
>>>> impacted submission stack.
>>>> 
>>>> For example, I was intending to contribute a very simple formula for
>>>> A099920 (Fibonacci-Lucas convolution): a(n) = a(n-1) + a(n-2) +
>>>> A000032(n-1) (Lucas numbers).
>>>> 
>>>> Though this one isn't earth-shattering, it still seems to me that any
>>>> simple formula for a sequence involving other core sequences might be
>>>> useful, and easy to review.
>>>> 
>>>> But I'm reluctant to post it under the current circumstances;
>> especially
>>>> since I have 3 proposed submissions already under review, including a
>> new
>>>> sequence posted 7 weeks ago with substantial activity on it.
>>>> 
>>>> So, are these types of submissions still OK?  Any rules-of-thumb to
>> keep
>>>> in mind before offering any more contributions?
>>>> 
>>>> Thanks,
>>>> Bob Selcoe
>>>> 
>>>> _______________________________________________
>>>> 
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>> 
>>> _______________________________________________
>>> 
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> 
>> 
>> 
>> --
>> Alonso del Arte
>> Author at SmashWords.com
>> <https://www.smashwords.com/profile/view/AlonsoDelarte>
>> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>> 
>> _______________________________________________
>> 
>> Seqfan Mailing list - http://list.seqfan.eu/
> 
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