@Michelle44 your calculations are correct, but the timing is a little off. You can contribute the $3975 up through April 15 of the following year. You do not need to make the contributions prior to Sept 30. But certainly the calculation of how much you can contribute is limited to the 9 months that you state in your case. See the difference?
It also presumes you are turning 65 in October. @KrisD15's response explains that,
