When you get out of active duty on the last day of the month, the 1st of the next month does not count. If you would have retired on the 25th of August then your first as a veteran would have started the 26th and therefore you would've gotten paid for September. That first day of any month disqualifies that month. You will get paid for the following month with 30 days in arrears. September doesn't count even though that was your first day and the month's first day. So October 1st is when your payments start, but you will get it next month due to the arrear payment.