Let a string of digits be called "good" if it satisfies one of the following properties: 7 It consists of a "4" followed by a good string. It consists of a "5" or "6" followed by a "7", "" or "9" then followed by a good string. It consists only of the digits 1, 2, and 3, such that there is exactly one 1 in the string It is the empty string. 7 Let an count the number of good strings of length n. (a) Construct a recurrence for an with initial conditions, explaining your reasoning. (b) Solve this recurrence. Let a string of digits be called "good" if it satisfies one of the following properties: 7 It consists of a "4" followed by a good string. It consists of a "5" or "6" followed by a "7", "" or "9" then followed by a good string. It consists only of the digits 1, 2, and 3, such that there is exactly one 1 in the string It is the empty string. 7 Let an count the number of good strings of length n. (a) Construct a recurrence for an with initial conditions, explaining your reasoning. (b) Solve this recurrence