Yes, you can use a fraction thereof.
The easiest thing (and this is the example provided in IRS Publication 969) is to calculate your annual HSA contribution limit as if you were under the HDHP all year, then multiply that limit by 9/12ths. So, ($3,450 + $1,000) times 9/12, which is $3,338 (rounded up).
NOTE: when you started Medicare, did the Social Security administration back-date your Medicare by 6 months (which they do for some taxpayers)? If so, then your Medicare started 6 months earlier than you might think. But perhaps you have already allowed for that.
Also, HDHP coverage depends on what coverage you had on the first day of the month. If you started a month with HDHP coverage and only later in the month, you started Medicare, then you still count that month as HDHP coverage.
Yes, you can use a fraction thereof.
The easiest thing (and this is the example provided in IRS Publication 969) is to calculate your annual HSA contribution limit as if you were under the HDHP all year, then multiply that limit by 9/12ths. So, ($3,450 + $1,000) times 9/12, which is $3,338 (rounded up).
NOTE: when you started Medicare, did the Social Security administration back-date your Medicare by 6 months (which they do for some taxpayers)? If so, then your Medicare started 6 months earlier than you might think. But perhaps you have already allowed for that.
Also, HDHP coverage depends on what coverage you had on the first day of the month. If you started a month with HDHP coverage and only later in the month, you started Medicare, then you still count that month as HDHP coverage.