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All but the very best flashes can not recycle atthat flash rate. For burst mode, you probably want to use focus modeAF-C or AF-A.
Page 73 in the manual; If the flash fires in L or H mode (pg. 65), only one picture will be taken each time the
shutter-release button is pressed. The shutterrelease may be briefly disabled to protect the flash after it has beenused for several consecutive shots. The flash can be used again after ashort pause.
The flash can take several seconds to recharge andduring that time the shutter is disabled. NR will also reduce thenumber of frames per second that you can take, and of course yourshutter speed has to be fast enough to allow for 4.5 fps.
In AF-S the shutter won’t fire unless the camera has achieved an in-focus state and the focus is locked.
In AF-C the camera continually focuses, but that isnot the same thing as being in focus. As you move the camera or yoursubject moves, the in focus state will come and go, but the camera willattempt to continuously re-focus. BUT, the shutter will fire even ifwhen the AF system has lost the in-focus state and the picture is outof focus.
In the D90 manual it indicates that the AF-A modewill only fire when the green in-focus light is on, just like in AF-S.The problem is that you probably won’t get 4.5 fps unless both thecamera and the subject are stationary and no time is lost between shotswhile the camera re-focuses.