[seqfan] Re: Augusto Santi's A351871

[Neil Sloane]

Robert,  Thanks for that analysis!  I will take the liberty of adding it
(slightly edited) to the sequence - I hope that is OK.
Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com



On Sun, Sep 18, 2022 at 11:18 AM 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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