Change A Date To Its Equivalent In Terms Of Quarters System

I have the formula where i can change a date to its equivalent in terms of quarters system with a date:

=CHOOSE(MATCH(MONTH(A2),{1,4,7,10}),"Winter","Spring","Summer","Fall")&YEAR(A2)

This is a school year configuration.


ex.
A2 = 10/1/2005: with the formula up there it turns into Fall 2005
i want to be able to add any number of years and the formula will still come up with the quarters system

also i would like A2 to be stationary and create a list of quarters for each year i add on

ex.
A2= 10/1/2005 B2=Fall 2005
B3=Winter 2006
B3=Spring 2006
B5=Summer 2006
B6=Fall 2006
etc.

If this is all possible lastly I would like to negate summer quarters