Is there non-recursive formula for the following sequence:
$$a_1=\frac12,$$
$$a_n=\frac12a_{n-1}^2+\frac12$$
If there is, how do you suggest I can determine it?
Asked
Active
Viewed 1,905 times
3
Barney Szabolcs
- 240
-
2What makes you think there is a non-recursive formula for that sequence? – Gerry Myerson Jul 23 '13 at 09:25
-
$a_n-a_{n-1}=(1/2)(a_{n-1}-1)^2$ but I'm not sure that helps. – Gerry Myerson Jul 23 '13 at 09:27
-
I've consulted with a mathematician who told me "ah let me think about it, it will not be too complicated" but then he did not have time to do it (in 5 minutes). And without the square it has a closed form. But I'll change my question. – Barney Szabolcs Jul 23 '13 at 09:42
-
1Never believe a mathematician. – Gerry Myerson Jul 23 '13 at 10:47
1 Answers
5
Making the change of variables $$a_n=1-2x_n$$ the recursion relation transforms into $$x_{n+1}=x_n(1-x_n).$$ This is a particular case of logistic map (with $r=1$). If it were solvable, I think it would be mentioned here along with $r=2,4$.
Start wearing purple
- 53,234
- 13
- 164
- 223