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{\bf MATH304  \hspace{1cm} HW 12\hfill \name}
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Due \textbf{Dec 1} before class, \textbf{answer without justification will receive 0 points}.
The typing the HW in \LaTeX{} is optional. 
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\question{}{
Solve the recurrence relation 
\[
 h_n = 3h_{n-2}-2h_{n-3}
\]
with initial values $h_0 = 1$,   $h_1 = 0$, and $h_2 = 0$ using characteristic equation.
Verify your answer for $h_3$.
}

\textbf{Solution:}


\question{}{
Solve the recurrence relation 
\[
 h_n = 3h_{n-2}-2h_{n-3}
\]
with initial values $h_0 = 1$,   $h_1 = 0$, and $h_2 = 0$ using generating functions.
}

\textbf{Solution:}



\question{}{
Solve the recurrence relation 
\[
 h_n = 3h_{n-1}+n^2-3
\]
with initial value $h_0 = 1$ using characteristic equation.
Verify your answer for $h_1$.
}

\textbf{Solution:}


\question{}{
Solve the recurrence relation 
\[
 h_n = 3h_{n-1}+n^2-3
\]
with initial value $h_0 = 1$ using generating functions. 
}

\textbf{Solution:}



\question{}{
Solve the recurrence relation 
\[
h_n = 2h_{n-1}+2^n
\]
with initial value $h_0 = 1$ using generating functions.  
Verify your answer for $h_1$.
}

\textbf{Solution:}


 
 

\question{}{
Let 
\[
 g(x) = \frac{x^2-3x+1}{4x^3-x^2-2x+1}
\]
be a generating function for a sequence $h_0,h_1,h_2,\ldots,h_n,\ldots$.
Find a linear homogeneous recurrence relation for the sequence and include also
the initial values.
}


\textbf{Solution:}






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