I have seen similar questions to this but they each seem to be special cases of this general question. Answering this would be beneficial to my research, but I am not a combinatorics expert, and this seemingly simple question eludes me. Is there a simple formula to calculate this? Everything I have seen online has been centered around things like "either 2 consecutive 1's or 0's" or "contains no ..".
If it helps, I know that for $m = 8$ bits and say the sequence is denoted $S(m,n)$ $$ S(m = 8, n = 1) = 255 \\ S(8,2) = 201 \\ S(8,3) = 107 \\ S(8,4) = 48 \\ S(8,5) = 20 \\ S(8,6) = 8 \\ S(8,7) = 3 \\ S(8,8) = 1 $$
Interestingly I'm finding that $S(8,4)=S(9,5)=S(10,6)=S(11,7)=48$ I haven't tested $S(12,8)$ because I don't want my computer to melt but I'm seeing a pattern... However this does not seem to work for $m<8$.