How to solve this recurrence relation related to the Group theoritic problem?

Asked Apr 19 '17 at 09:23

Active Apr 19 '17 at 09:39

Viewed 103 times

I was trying to count the number of elements of order 2 in the group $S_n$ and somehow translated the problem to the following recurrence relation.

$a_{n+1} = a_n + na_{n-1}$ with the initial conditions $a_0=0, a_1=1, a_2=2$ for $n \geq 3$

I don't know how to proceed after this stage. Kindly help.

edited Apr 19 '17 at 09:39

asked Apr 19 '17 at 09:23

joy

Google for "linear recursion", or search on MSE - e.g., here. – Dietrich Burde Apr 19 '17 at 09:27
Also the number of elements of order $2$ in $S_n$ have been determined on MSE. For a start, read this question, and this one. – Dietrich Burde Apr 19 '17 at 09:28
the solution is not so simple, see WolframAlpha – Dr. Sonnhard Graubner Apr 19 '17 at 09:35
1

Sorry, add the condition $n \geq 3$ – joy Apr 19 '17 at 09:39
@ Dr. Sonnhard Graubner, kindly provide me the link. – joy Apr 19 '17 at 09:40
@joy Isn't it $n\ge 2$, otherwise, how do we get $a_3$ ? – Peter Apr 19 '17 at 09:50
https://www.wolframalpha.com/input/?i=a(1)%3D1+,+a(2)%3D2+,+a(n%2B1)%3Da(n)%2Bna(n-1) – Peter Apr 19 '17 at 09:54
@Peter I am sorry but I don't call "solution" the output given by WA with this input... – Jean Marie Apr 19 '17 at 10:06
@JeanMarie Apparantly, Wolfram Alpha does not find a solution, maybe there is no closed form. I posted the link because joy was interested in the output. – Peter Apr 19 '17 at 10:26

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