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Find a functional equation and solve it for sequence of generating functions whose coefficients satisfy (assume and =1):

1. Find a functional equation and solve it for sequence of generating functions whose coefficients satisfy (assume   and  =1):
  1.  
2. Find a recurrence relation and associated generating function for the number of n-digit ternary sequence that have the pattern "012" occurring for the first time at end of the sequence.
3. Find and solve simultaneous recurrence relation for determining the number of n-digit ternary sequences whose sum of digits is a multiple of 3
Please see the attached file for the fully formatted problems.

=1):
2. Find a recurrence relation and associated generating function for the
number of n-digit ternary sequence that have the pattern “012”
occurring for the first time at end of the sequence.
3. Find and solve simultaneous recurrence relation for determining the
number of n-digit ternary sequences whose sum of digits is a multiple of
3

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