Ps3 controller not syncing

From experience it does sound like the display output is set wrong. The above information is a tad bit wrong, because you do not go into recovery mode to reset the display.

Try this it may work it may not, but it's worth trying it the right way.

1. Take your controller, turn it over, stick a pencil lead or pin or something down into the reset hole, and press the button.
2. Hook up the controller to the system via the supplied usb cable.
3. Press the PS (Home) button on the control or turn on the system via the power button.

This should fix your controller issue which you need to fix before trying to reset the display correctly. If the controller has any sort of surge or anything when charging it will do this, and it happens a lot when you use something besides the console to charge the controller. It is basically just resetting and pairing the controller again.

If the controller reset works, and the lights stop blinking turn off the console via holding down the power button. Do not unhook the cord.

1. Remove your hdmi or av cable from the console. If it's hdmi remove it from the tv as well, cut the tv on, and back off.
2. tightly plug in the hdmi or av cable to both the console and tv. Move the cable so that it's not coiled up. HDMI cables will break pretty easily.
3. Press and hold the power button on the console until it beeps for a second time. This is display reset mode, and not recovery mode.
4. IF the display comes up cut on the controller press left then X.

I know this sounds almost like the same thing posted above, but again that's saying go into recovery mode, which is the wrong place to be. Also, if the hard drive is corrupted it will pop up on screen and tell you that you need to fix it. It won't mess up your video settings, because it will automatically set to the lowest quality for your video connection.

To me seems like controller is either dead, or for some reason incompatible.

You said that you bought new PS3, is controller that comes with PS3 working, and the one that you have from old console (?) not working? Is the contoller that is not working an original Sony controller, or a compatible one from some other brand?