2. Decimal Strings

A dec-string is a sequence of characters from the 10-character alphabet {0,1,2,3,4,5,6,7,8,9}. For example, these are dec-strings:

0 36562342320 49548362729 

Let n0 be an integer.

1. What is the number of dec-strings d1,,dn of length n4 such that d1d200 or d3d411.
2. A dec-string d1,,dn is bad if di=di+1 or di+di+1=9 for at least one i{1,,n1} and it is good otherwise. What is the number of good dec-strings of length n?
3. A dec-string d1,,dn is 2-bad if, di=dj or di+dj=9 for some i2-good otherwise. What is the number of 2-good dec-strings?
4. A dec-string d1,,dn is k-bad if di=dj or di+dj=9 for some ik-good otherwise. What is the number of k-good dec-strings of length n?