[seqfan] Re: Augusto Santi's A351871

[Neil Sloane]

PS re A351871: Here is my hand-wavy explanation for R,
Gerbicz's observation.  In order for the sequence to cycle there has to be
a gcd that is big enough to shock the sequence down to a much smaller
value.  If the two starting values are odd then the gcd can't exceed
one-third of the sum, and that is not enough of a shock, and the sequence
keeps growing. In the other case the gcd can be half the sum, and then
a(n+1) = (a(n)+a(n-1))/2  + 2, and that is enough to reset the sequence so
that it has a chance to cycle.



On Tue, Sep 20, 2022 at 1:50 AM Neil Sloane <njasloane at gmail.com> wrote:

> Robert Gerbicz made some very interesting discoveries and conjectures
> which are now part of A351871.
>
> Let me encourage people who have more comments to post them to this
> mailing list, so everyone can see them.
>
> I was hoping that when one compared the graphs of these 4 sequences
> A351871 & A355914; A355898 & A355899,  it would reveal some profound truth,
> but I don't see it.
>
> Another suggestion: Let a(n) = A351871(n).  Plot the successive points in
> the plane given by (a(1),a(2)), (a(2),a(3)),  (a(3),a(4)), ... What does
> their trajectory look like? Ditto for A355898.
> This was a device used with great success by Remy Sigrist in explaining
> the "Breche de Roland" sequence from earlier this summer.
> Best regards
> Neil
>
> Neil J. A. Sloane, Chairman, OEIS Foundation.
> Also Visiting Scientist, Math. Dept., Rutgers University,
> Email: njasloane at gmail.com
>
>