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Define the linear recurrence $a_n=F_na_{n-1}+a_{n-2}$ with $a_0=1$, $a_1=2$, and $F_n$ being the Fibonacci series ($F_0=F_1=1$). Is there a possible closed form for $a_n$, or even a series representation? I've looked at several resources, but many of them just deal with recurrences with constant coefficients.

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  • $\begingroup$ Check again your recursive formula. $\endgroup$ – hamam_Abdallah Dec 27 '16 at 15:10
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    $\begingroup$ Is the second term really $an$, or did you mean $a_n$, or even $a_{n-2}$? $\endgroup$ – J. M. is a poor mathematician Dec 27 '16 at 15:10
  • $\begingroup$ I have often seen two definitions of Fibonacci series. Either $F_0=0$ and $F_1=1$ or $F_0=F_1=1$. Which definition are you using here? $\endgroup$ – Noble Mushtak Dec 27 '16 at 15:13
  • $\begingroup$ @J.M.isn'tamathematician Sorry, forgot an underscore $\endgroup$ – LegionMammal978 Dec 27 '16 at 15:24
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    $\begingroup$ This is A026822 in OEIS. There's not much more information there. $\endgroup$ – rogerl Dec 27 '16 at 15:32

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