it could mean the power isnt charging the battery just running the laptop. some laptops can happily run off the mains, while others constantly drain the battery and constantly charge it. so it could be your battery is fully charged and being used by initial boot, but because your connected to the house power the battery gets a small surge to replace what it just used and then stops charging. to test this. remove the charger and run off the battery for half an hour. then just plug the charger back in. the battery should start charging but now the icon should start moving also...
if it does this, it means your laptop has battery saving features that should extend the life to over double what they normally last...