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Suppose we want to find an explicit formula for
the sequence satisfying , ,
| for .
Since satisfies a linear recurrence with characteristic polynomial
we know that there exist constants and
for all .
The formula (7) is called the general solution
to the linear recurrence (6).
To find the particular solution with the correct values of and ,
we use the known values of and :
Solving this system of equations yields and .
Thus the particular solution is
(As a check, one can try plugging in or .)