Did the Roman emperors use the ascension reckoning or non-ascension reckoning when counting their reigns?

For instance, Emperor Tiberius ascended the throne on September 18th, 14 A.D. Using ascension reckoning he would have to wait until January 1st before being allowed to count his reign, while if he used the non-ascension method, he could began counting his reign immediately.Did the Roman emperors use the ascension reckoning or non-ascension reckoning when counting their reigns?
The sites just say from the beginning of the reign, but Rome also used AUC - from the founding of the city.
So it might be in the second year of the reign of Consul/Emperor......................, 365AUC.

http://calendopedia.com/counting.htm

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