I'm trying to figure out how to find closed form equations for recurrence relations. I can find lots of examples for solving equations such as $a_{n} = ca_{n-1} + ca_{n-2}$ and $a_{n+1} = ca_{n} + ca_{n-1}$, but I came across a problem of $a_{n+2} = ca_{n+1} + ca_{n}$. How would the typical method for finding closed form equations (like the one described here) change for this problem?
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2 Answers
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All three equations you give are (essentially) the same. For example, put $n+2= m$ to reduce the third to the first.
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Set $b_n=a_{n+2}$ or $d_n=a_{n+1}$ and solve for $b_n$ or $d_n$
answered Aug 15, 2013 at 8:11