Closed form of $a_{n+1}=\sqrt{a_n+1}, a_1=1$

Question

I would like to find the general term, closed form or analytic solution for the recurrence relation $a_{n+1}=\sqrt{a_n+1},a_1=1$.

I looked around the internet but could not find much about this problem. If anyone knows any paper or research about this problem, I would like to know.

My research:

• The sequence is a bounded increasing sequence and converges to $\frac{1+\sqrt{5}}{2}$.

• The sequence is contractive.

• A similar problem $a_{n+1}=\sqrt{a_n+2}, a_1=\sqrt{2}$ has a closed form $a_n=2\cos{\frac{\pi}{2^{n+1}}}$.

• A similar problem $a_{n+1}=a_{n}^2+c$ does not(?) have a simple solution except for $c=0,-2$ (I am not sure if I understand what's written here)

The 3rd website and 4th website seems to be related since the equation $a_{n+1}=a_{n}^2+c$ turns into $a_n=\sqrt{a_{n+1}-c}$ and if we swap $a_n,a_{n+1}$ and set $c=-2$(which is one of the values mentioned) we get $a_{n+1}=\sqrt{a_n+2}$. However, I do not know how to make use of this connection.

Answer 1

This may not be exactly what you are looking for but it is close.

Define the constants $$ s:=\sqrt{5},\; g:=(1+s)/2,\text{ and } c\approx -2.1972839287883129714.$$ Define using power series the function

$$ T(x) := x + \frac{5 - s}{20} x^2 + \frac{5 s - 11}{20} x^3 + \frac{497 s - 1110}{3800} x^4 + \cdots. $$

Define the sequences

$$ b_n := T(c\cdot(2g)^{-n}), \qquad a_n := g+b_n. $$

Note that $\,\lim_{n\to\infty} b_n = 0\,$ and $\,\lim_{n\to\infty} a_n = g.\,$ The sequence $\,a_n\,$ satisfies

$$ a_{n+1}=\sqrt{a_n+1},\qquad a_1=1. $$

Unfortunately, I can only determine the coefficients of $\,T(x)\,$ one by one and have no closed form for the function itself or its coefficients.

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Some more details are here. The idea is to ensure that

$$ g \!+\! T(x) \!=\! \sqrt{g^2 \!+\! T(2gx)},\; \text{ or }\;T(x)^2\!=\! T(2gx) \!-\! 2gT(x). $$

Given the definition of $\,a_n\,$ this implies that $\,a_{n+1} = \sqrt{a_n+1}.\,$

Use the method of undetermined coefficients, to solve for the coefficients of $\,T(x)\,$ one by one. For example, assume that $\, T(x) = x + u_2x^2 + O(x^3).\,$ Then use $\, x^2 + O(x^3) = (2gx + u_2(2gx)^2) - (2gx + 2gu_2x^2) = (5+s)u_2x^2\,$ to solve for $\,u_2.$

The value of $\,c\,$ is determined as the unique value that yields $\,a_1 = 1.\,$ Also, it is the limit $\,c = \lim_{n\to\infty} b_n(2g)^n.\,$