Secondary trigger signal creation

Sounds like a job for a 555 timer, which are very cheap. I have only recently started playing with them, but one of the engineers here will probably come up with a circuit shortly. Until then you could google:
"555 delay timer circuit" and "555 timer calculator".

Yes, the 555 sounds like an excellent solution for this. It triggers on a falling edge and can drive up to 200ma (if the coil on the relay that you're using requires more than 200ma, then you'll, also, need to add a driver to the output of the 555.

Is there a case where the sensor signal might occur again before the 1-3second cycle completes? And, if it does, what should be the outcome?

It is not just a relay, is a relay module for an Arduino so the current drain should should be OK,
I'm not sure, the data-sheet for my sensor is likely Google translated from Chinese, and although there is a repeat jumper there is no instructions for it
I'm uploading the data-sheet.
Thanks for the help

Hi,

I have a twin relay board, one relay I trigger with a sensor signal V1=3.5V.
I want to activate the second relay with a signal V2 that goes high for about 1-3 secs (settable) after V1 goes low.
I was thinking if is possible to do it with few analog components, as I want to keep it as simple and cheap as possible.
Any help with a possible circuit would be appreciated.

Many thanks,
james

P.S.
What I am trying to do, is to detect the presence of a person in a room, then with relay1 turn some lights on, and keep them on whilst the room is occupied.
When the room is empty, use the sensor's timer to wait for 3-10 minutes, then turn the lights off (relay1), and activate relay2 to sound a buzzer/mp3 for 1-3 secs.
I hope this helps.

pir sensor datasheet