Nesse caso, o manual pode ser muito útil.
Suponho que você queira algo como: PROMPT='$HOME@%M:%3~ '
Aparentemente, na verdade, ler as 4 frases na pequena página de manual é muito trabalhoso. Talvez porque seja um link:
13. Prompt Expansion
13.1 Expansion of Prompt Sequences
Prompt sequences undergo a special form of expansion.
This type of expansion is also available using the -P option
to the print builtin.
If the PROMPT_SUBST option is set, ...
Certain escape sequences may be recognised in the prompt string.
If the PROMPT_BANG option is set, ...
If the PROMPT_PERCENT option is set, ...
13.2 Simple Prompt Escapes
13.2.1 Special characters
%%
A '%'.
%)
A ')'.
13.2.2 Login information
%l
The line (tty) ...
%M
The full machine hostname.
%m
The hostname up to the first '.'. An integer may follow the '%' to specify
how many components of the hostname are desired. With a negative integer,
trailing components of the hostname are shown.
%n
$USERNAME.
%y
The line (tty) ...
13.2.3 Shell state
%#
A '#' if the shell is running with privileges, ...
%?
The return status ...
%_
The status of the parser, ...
%d
%/
Present working directory ($PWD). If an integer follows the '%', it
specifies a number of trailing components of $PWD to show; zero means
the whole path. A negative integer specifies leading components,
i.e. %-1d specifies the first component.
%~
As %d and %/, but if $PWD has a named directory as its prefix, that part
is replaced by a '~' followed by the name of the directory. If it starts
with $HOME, that part is replaced by a '~'.
%h
%!
Current history event number.
%i
The line number ...
%I
The line number ...
%j
The number of jobs.
%L
The current value of $SHLVL.
%N
The name of the script, ...
%x
The name of the file ...
%c
%.
%C
Trailing component ...These are deprecated... equivalent to %1~ and %1/ ...
13.2.4 Date and time ... Ignored ...
13.2.5 Visual effects ... Ignored ...
13.3 Conditional Substrings in Prompts ... Possibly useful but ignored ...