[seqfan] Re: Proof For A269254
njasloane at gmail.com
Mon Oct 23 06:12:59 CEST 2017
The missing steps in the proof can be filled in like this.
For example, we know by construction that b(n) satisfies a second-order
and c(n) a third-order recurrence. We want to prove that the componentwise
product b(n)*c(n) satisfies a second order recurrence. Well, a basic
theorem about linear recurrences tells us that the product satisfies a
linear recurrence of order at most 2*3 = 6. And NOW we can use Gfun in
Maple to find it,
and it turns out to satisfy a second-order recurrence. All that gfun does
in a case like this is to make a call to Maple's convert[ratpoly], which
uses Pade approximations. I assume this can be made rigorous if anyone has
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 Sun, Oct 22, 2017 at 10:38 PM, Neil Sloane <njasloane at gmail.com> wrote:
> The case 110 is certainly always composite. I didn't work out all the
> details, but, first, a(3n+1) is zero mod 3 (just run the recurrence mod 3)
> Second, it seems that a(3n) = b(n)*c(n),
> where Don Reble found b(n) = 110*b(n-1)-b(n-2), and using gfun I get
> c(n) = 12099*c(n-1)-12099*c(n-2)+c(n-3)
> Third, similarly, it seems that a(3n+2) = d(n)*e(n), where d(n) and e(n)
> satisfy the same recurrences as b(n) and c(n), except with different
> initial conditions.
> This may not be a legal proof, but one can now use gfun to verify that
> a(3n) and b(n)*c(n) satisfy the recurrence
> A(n) = 1330670*A(n-1) - A(n-2),
> and so must be equal.
> Likewise, a(3n+2), d(n)*e(n) satisfy the same recurrence
> (with different initial conditions)
> and so are also equal.
> Best regards
> 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 <(732)%20828-6098>; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
> On Sun, Oct 22, 2017 at 5:49 PM, L. Edson Jeffery <lejeffery2 at gmail.com>
>> Nice proof. I just submitted the array used for computing A269254. Here is
>> the draft if anyone wants to add to it:
>> I also stated the theorem in A269254 and gave a link to your proof on
>> Meanwhile, a(110) is still unknown. My Mathematica program ran all night,
>> and is still running, to no avail.
>> Ed Jeffery
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