No, there are no ongoing patterns. There are some micro-patterns inside pi (such as 123456789) but they have no meaning.
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Pi
pi is an irrational number. It can not be expressed as a ratio of two integers.
Pi doesn't end. It has been computed as of October 2011 to ten trillion places with no discernible pattern.
Infinitely many. pi is not just an irrational number but a transcendental number. All irrational numbers have infinite decimals that do not go into a recurring pattern.
22 divided by 7 * * * * * That is an APPROXIMATE value of pi. The exact value cannot be calculated since pi is a transcendental number - a special kind of irrational number. It has an infinite decimal representation with no recurring pattern. That would be true in any base - binary, octal, or another base (other than pi itself, or a power of pi).