x->x*ceiling(x)

[John Conway]

On Thu, 29 Aug 2002, David Boyd wrote:

> In answer to John's question, the result about 14,23 is in
> my paper in Acta Arith, v. 24 (1977), 89--98.   Shortly
> thereafter, I was able to show that starting with any of
> the following gives a Pisot sequence which is not linearly
> recurrent:  7,15   8,14   9,15  9,17  9,19  10,18  10,21.
> David Cantor suggested that probably 4,13  also is a bad
> starting pair.   One of my old Macs did a computation
> verifying this but since it ran for a few months I wouldn't
> claim that this has yet been proved, but if true it is
> a matter of a finite computation.

   Could you briefly explain how you prove such things
(and in particular, why they reduce to finite computations)?

> I don't think the sequences generated by  x -> x*ceil(x)
> have much to do with Pisot sequences....

   I didn't think there was much real resemblance either -
just a reminiscence, so to speak.  

   Regards, John Conway