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Find the formula of the sequence $$u_1=a, u_{n+1}=u_n^2+u_n$$

Is there a "simple" formula for the sequence ?

Math Lover
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RopuToran
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    This is called a quadratic map. The standard form comes from making the substitution $u_n = v_n - 1/2$ to get $v_{n+1} = v_n{}^2 + 1/4$ – DanielV Sep 13 '18 at 17:55

1 Answers1

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There is no simple formula for $u_n$ in terms of $a$. Indeed, $u_n$ is a polynomial expression on $a$ of degree $2^{n-1}$.

If you seek a simple formula for $u_n$ to understand its convergence, then this is hopeless because convergence is explained by Julia sets, which are quite complicated. In this case, the Julia set of $f(z)=z^2+z$:

enter image description here

(Image computed with WA)

lhf
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