There are problems where we need to find an explicit formula for some recursive sequence such as the Fibonacci sequence, defined by $f_0=0, f_1=1,f_n=f_{n-1}+f_{n-2}$ where we can find $f_n$ in terms of $n$ only, or another example is $a_1=1,a_2=2,a_n=2a_{n-1}+3a_{n-2}$ where we can find $a_n$ in terms of $a_1,a_2,n$. I'm wondering if it is possible to find an explicit function for any given recursive sequence, in terms of at least one term of the set $\{a_1,a_2,...,a_k,n\}$.
Is it possible to find an explicit function for any given recursive sequence?
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linear-algebra
sequences-and-series
recursion
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1Are you specifically interested in linear recurrences? If so, then yes. See, e.g., [this](http://mathworld.wolfram.com/LinearRecurrenceEquation.html) for the examples you discuss. These methods can be generalized (but the linearity is crucial). – 2017-01-04
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0@lulu thanks for the link! – 2017-01-04
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0@lulu thanks for the link as well. I'll probably have a better look at these in the future. – 2017-01-04