[seqfan] Re: Question about submission guidelines

[Vladimir Shevelev]

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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