Sequences of two letters

[Apus Chuh]

Consider a sequence of two letters, a and b, of length m+1.

Define s := #(consecutive 'ab' or 'ba' pairs) - #(consecutive 'aa'
pairs)

where the first term is therefore equal to #(b in all but the first
position). How many sequences have s > 0, as a function of m? Can I
upper bound this number in terms of m?

[Chip Eastham]

Perhaps I don't fully understand the question:

If a sequence consists solely of b characters, then
it seems both terms that define s are zero, so the
observation "the first term is therefore equal to
#(b in all but the first position position)" needs a
bit of refinement.

Also you want to count only "consecutive" pairs of a
certain form. What happens when the consecutiveness
stops? Do you count only the longest streak? Do you
add the lengths of streaks? Count only the final one,
i.e. start the count over at zero when a streak ends?

regards, chip