K-sequence/ -LN(gFRAC( x)), Take-it!.

[Don McDonald]

In message <01ce01c2f98b$d7d31320$418239d2 at computer> you write:
seqfans,

About 1976 I generated (iterated) random reals on
Hewlett-Packard HP25  programmable keystroke palm calculator. by..

x(n)= -LN(GFRAC( x(n-1) ).
ln() is log natural
gFrac() is the fractional part of a Real. = x -floor(x).

[the implied floor(x) values are an integer sequence.]

The game was to seize the larger occurences of x(n)
as they flashed on the display (and hopefully compile a
large score within one minute limit)
and pass on small or average x(n)s. (The game incorporates
speed features, e.g. waiting time delay thinking is penalised.)

I have also done the exercise on BBC-Acorn computers.

The sequence must repeat, because there are only
finitely many machine numbers.

It is a very simple non-uniform random number generator.
And I can see large values several steps before they display.

What is the length of the cycle/s.? (millions.)
What is the distribution of x(n)s?
Depends on rounding.

A starter value can be x(0)= SQRT(1234...) , etc.

Experiment, x(0)= SQRT(1+.1*n)).  gives 9 trial sequences.

(I thought what is a 'strange-attractor?')

don.
mcdonald
05.04.03  00:15

>     Hello, seqfans.
>     I conjectured the following sequence is divergent.
> 
> >   x(n)=[a*x(n-1)+b]/p^r,
> >   a, b are real number, [x] is integer part of x,
> >   p is prime, p^r is the highest power of p dividing [a*x(n-1)+b]
> 
>     x(0), p, a, b = 107, 2, 1.6, 1.1
>
.Calc.Profile.eisintegsq.Seqfan.Take_it_n