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$x_1=1$, $x_2=2$, $x_n=n\cdot x_{n-1}+\frac{n(n-1)}{2}x_{n-2}$

Thanks to Barry Cipra's suggestion, now I have obtained the solution for $x_n$:

$x_n=\frac{n!}{\sqrt{3}}((\frac{1+\sqrt{3}}{2})^n-(\frac{1-\sqrt{3}}{2})^n)$

Chaos Mage
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1 Answers1

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Hint: Let $u_n=x_n/n!$. Find (and solve) a recurrence relation for $u_n$.

Barry Cipra
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  • Brilliant! It is much simplified. Then it is deduced to solve this problem: $u_1=1$, $u_2=1$, $u_n=u_{n-1}+\frac{u_{n-2}}{2}$.

    But I still can not find a close form solution for it.

     – Chaos Mage Feb 11 '14 at 09:32
  • Ok, I find a solution for $u_n=u_{n-1}+\frac{u_{n-2}}{2}$ in http://en.wikipedia.org/wiki/Recurrence_relation#General_methods – Chaos Mage Feb 11 '14 at 10:55
  • But what if $nx_n=x_{n-1}+x_{n-2}$ – Nirvana Jul 31 '14 at 11:43
  • @Rejo_Slash, that's a completely different question. If you post it as such, you will probably get an answer. – Barry Cipra Jul 31 '14 at 12:38
  • I posted it..But couldnt end up in a solution...I just saw your methods..Just checking whether you know how to deal it..Please go through the link of my post! – Nirvana Jul 31 '14 at 13:03