[seqfan] Re: Augusto Santi's A351871

[Robert Gerbicz]

Hi !

Really not searched previously for this, just because it is slightly harder
to check this also, and in the gmp library there is no map/set. Basically
it is not that surprising since we have very few cycles, just doing a basic
search I can see only two cycles for length=3 and unique for other lengths.
For length=3: the cycles are 27,32,60 and 32,27,60, as expected since for
this length every 3 different number's permutation is giving a cycle.
Just for completeness:
For a(1)=50, a(2)=57 it goes to a cycle with length=9.
For a(1)=4, a(2)=2 it goes to a cycle with length=155.


Robert Dougherty-Bliss <robert.w.bliss at gmail.com> ezt írta (időpont: 2022.
szept. 18., V, 19:47):

> Dear Robert,
>
> I ran some experiments which indicated that, often, initial conditions
> leading to the same period also led to the same periodic values. Was this
> true in your experiments as well?
>
> Robert
>
>
> On Sun, Sep 18, 2022, 11:18 Robert Gerbicz <robert.gerbicz at gmail.com>
> wrote:
>
> > a(1), a(2) are the first two (positive) integers then for n>2:
> > a(n)=g+(a(n-1)+(a-2))/g, where g=gcd(a(n-1),a(n-2)).
> >
> > If a(1) and a(2) are odd then easily all numbers in the sequence are odd.
> > If a(1) or a(2) is even, then with induction for every two consecutive
> > numbers in the sequence at least one of them is even.
> > It gives that it is a partitioning of the sequences into 2 groups. And
> what
> > is interesting is that it looks like that in the first group the sequence
> > always goes to infinity, and in the second group it always goes to a
> cycle.
> > At least I have no counter-example for this conjecture.
> >
> > And three more cycle lengths:
> > for a(1)=52, a(2)=378 the sequence starts with:
> > 52, 378, 217, 92, 310, 203, 514, 718, 618, 670, 646, 660, 655, 268, 924,
> > 302, 615, 918, 514, 718, ...
> > It gives a cycle length of 12, starting at 514.
> >
> > for a(1)=264, a(2)=1037 the sequence starts with:
> > 264, 1037, 1302, 2340, 613, 2954, 3568, 3263, 6832, 10096, 1074, 5587,
> > 6662, 12250, 9458, 10856, 10159, 21016, 31176, 6532, 9431, 15964, 25396,
> > 10344, 8939, 19284, 28224, 3971, 32196, 36168, 5709, 1302, 2340, ...
> > It gives a cycle length of 29, starting at 1302.
> >
> > for a(1)=542, a(2)=6017 it gives a cycle of length 802, the maximum term
> > is 557981456058.
> >
> > --
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> >
>
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