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?
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2$\begingroup$ What makes you think there is a non-recursive formula for that sequence? $\endgroup$– Gerry MyersonJul 23, 2013 at 9:25
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$\begingroup$ $a_n-a_{n-1}=(1/2)(a_{n-1}-1)^2$ but I'm not sure that helps. $\endgroup$– Gerry MyersonJul 23, 2013 at 9:27
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$\begingroup$ 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. $\endgroup$– Barney SzabolcsJul 23, 2013 at 9:42
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1$\begingroup$ Never believe a mathematician. $\endgroup$– Gerry MyersonJul 23, 2013 at 10:47
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1 Answer
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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 $r=2,4$.
answered Jul 23, 2013 at 9:55