I came across a very peculiar recurrence relation :

$\sqrt {T(n)} = \sqrt {T(n-1)} + 2 \sqrt {T(n-2)} $

with initial values $T(0) = T(1) = 1$

Any helps on how to find it

  • 2
    $\begingroup$ Consider denoting $f(n)=\sqrt{T(n)}$, and solving the linear recurrence relation for $f(n)$. $\endgroup$
    – Ariel
    Sep 2, 2015 at 14:19
  • $\begingroup$ let me try that $\endgroup$ Sep 2, 2015 at 14:22

1 Answer 1


Let $f(n)=\sqrt{T(n)}$, $f$ satisfies the linear recurrence relation $f(n)=f(n-1)+2f(n-2)$. The characteristic polynomial is $x^2-x-2$ and it's roots are $-1,2$, so $f(n)$ is a linear combination of $(-1)^n,2^n$. After substituting the initial conditions you get $f(n)=\frac{1}{3}(-1)^n+\frac{2}{3}2^n$.


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