This 1984 paper reviewing various machines' RND functions states that:
In a study of the statistical and temporal characteristics of random
number generators for microcomputers, several peculiarities related to
seeding random number generators were noted. Some vendor-supplied
generators, including those on the Apple II+, Apple IIe, Osborne
Executive, and IBM-PC, were so seriously flawed that they should not
be used for simulation studies.
The same authors wrote an article for BYTE magazine in 1987 - volume 12, number 1, page 175 - where they mention one of the RND cycles having a period of only 202 values. The problem can be easily seen in this handy YouTube video where the RND characteristics are not sufficient to plot every pixel on a hires screen.
In Apple Assembly Line Volume 4 Issue 8 Bob Sander-Cederlof documented the RND problems and presented his own smaller version of the original Call APPLE algorithm as a USR function.
The RND function in Applesoft is faulty, and many periodicals have
loudly proclaimed its faults. "Call APPLE", Jan 83, pages 29-34, tells
them in "RND is Fatally Flawed", and presents an alternative routine
which can be called with the USR function.
If you want to see some non-random features using RND, type in and RUN
the following program:
30 HPLOT X,Y
40 GO TO 20
You will see the Hi-Res screen being sprinkled with dots. After about seven minutes, but long before the screen is full,
new dots stop appearing. RND has looped, and is replotting the same
sequence of numbers. Another test disclosed that the repetition starts
at the 37,758th "random" number.
I re-implemented the Call APPLE algorithm, and my listing follows. The
Call APPLE version would not quite fit in page 3, but mine does with a
little room to spare.
The listing is in S-C Assembler format. The assembler is available and I had a quick look, but I don't know how to use it. At some point I'll get around to converting this and post the hex here.