I have a project where I know a single computer and a single printer will be the only things on the network. What I want to do is detect when the printer is connected to the network. I also know that the computer is 192.168.3.1. However, with DHCP I won't know the printer address (yes, it could be made static to make it easier but, 'they' don't like that. 'They' want it dynamic)

What I have is a script that does the following and it works.

nmap -sP 192.168.3.0/24 
 | awk '/192.168.3/ && !/192.168.3.1$/' 
 | sed 's/Nmap scan report for //'

Nmap output

 Nmap scan report for 192.168.3.1
 Host is up (0.014s latency).
 Nmap scan report for 192.168.3.100
 Host is up (0.012s latency).
 Nmap done: 256 IP addresses (2 hosts up) scanned in 2.54 seconds

Script output

 192.168.3.100

It only takes a couple seconds to work but is there a better/cleaner/faster way?

There's no need to scan the entire subnet if you know that you're not interested in part of it. (Avoiding the computer means you don't need to discard its result.)

nmap -oG - -sn 192.168.3.2-254 | awk '$NF=="Up" print $2'

or if you prefer using the XML output instead of the grep output

nmap -oX - -sP 192.168.3.2-254 | xmlstarlet sel -t -m '//address[@addrtype="ipv4"]' -v '@addr' -n

Use -sP instead of the newer -sn if your version of nmap requires it.

Incidentally, although your system administrators may want you to have your printer on DHCP, there should be little reason why they can't arrange for it to have a known unchanging address. (I do that for printers on my networks so that printer software doesn't need to worry about IP addresses changing unexpectedly.) Sometimes this is known as a "sticky" address, to differentiate it from a static (non-DHCP) address or a pseudo-random dynamic (DHCP) address.

Are you sure the DHCP server itself won't be on your subnet? Otherwise, how is your printer going to get its dynamic address?

You can accomplish this with the following awk command:

nmap -sP 192.168.3.0/24 
 | awk '/192.168.3/ && !/192.168.3.1$/print $NF'

This is telling awk to print the last field of the matched line(s)

Perhaps an even more efficient solution using only grep (requires GNU grep or a grep that supports perl regex):

nmap -sP 192.168.3.0/24 
 | grep -o -P '192.168.3.(?!1$)[0-9]+'

This is greping for -o (only) the IP address matching any IP beginning with 192.168.3 except for 192.168.3.1

Once you have the IP address for the printer you can then get the host name with nmblookup -A 'IP address'. This should help in the future if the IP changes since you'll be asking the network for a machine with this name vs trying to find a machine by IP address.