Final answer:
To find the recurrence relation for f(n), we analyze cases based on the last characters of (n-1)-digit strings to ensure the 'tag' substring does not form when a new character is added, leading to a sum of different contributions based on strings ending with 'ta', 't', not 'ta', and not 't'.
Step-by-step explanation:
The student is asking for a recurrence relation for the function f(n), which denotes the number of n-digit strings made from the characters a, c, t, g that do not include the substring tag. To establish this recurrence relation, we must consider strings of length n ending in different characters and ensure that adding another character does not create the forbidden substring 'tag'.
Let's break the problem down:
- An n-digit string not ending with 'ta' can be followed by any of the four characters, thus contributing 4*f(n-1) to f(n).
- An n-digit string ending with 'a' but not with 'ta' can be safely followed by 'c', 'g', or another 'a' — contributing 3 times the number of (n-1)-strings ending with 'a' but not 'ta' to f(n).
- An n-digit string ending with 't' but not 'ta' can be followed by 'c', 'g', or 'a' (which doesn't form the 'ta' at the end) — contributing 3 times the number of (n-1)-strings ending with 't,' but not 'ta,' to f(n).
- For the n-digit strings already ending with 'ta,' we can only add 'c' or 'g,' not to form 'tag' — contributing 2 times the number of (n-1)-strings ending with 'ta' to f(n).
From these points, we derive a recurrence relation that accounts for all cases without creating the substring 'tag'. The detailed formation of this recurrence requires an understanding of combinatorial mathematics and careful enumeration of possibilities.