# $35.2850899…$ has a closed form ??

Consider the function $$t(x)$$ defined as :

$$x_1 = x$$ $$x_2 = x$$ $$x_3 = 2 x^2$$ $$x_4 = 4 x^4 + 2 x^2$$

and for $$n > 4$$

$$x_{n} = \frac { x_{n-1}^2 + x_{n-2}^2 + x_{n-3}^2}{x_{n-2} + x_{n-3} + x_{n-4} }$$

If the sequence converges to a constant then we define $$t(x) = \lim x_n$$.

This function is not so well understood by me.

I assume it is analytic for instance. But i have no formal proof.

The function grows fast. But I am not sure how fast exactly. ( ofcourse 10 iterations give a good asymptotic and then notice it goes double exponentially fast to its lim )

I have no series expansion , integral representation , differential equation , continued fraction or such for this. Nor do I have a way to compute these values in a different way such as different iterations , a koenigs type formula , combinatorical methods , fractals , cellular automatons , a bifurcation point , the area of a filled julia set etc etc.

practically there is not a big problem for small imput since it converges rapidly. But the values are mysterious to me.

In particular is $$t(1)$$ transcendental ?? Can we even prove it to be irrational ??

Do we have a closed form for $$t(1)$$ ? [ main question ! ]

This all reminds me of the Somos sequences and the Somos constant ( which has a closed form as the derivative of a lerch form ! )

Assuming it is analytic , how does its analytic continuation look like ? Is analytic continuation possible to any complex number or its neighbourhood , or is there a natural boundary ?

numerically we have

$$t(1) = 35.2850889...$$

Do you recognize this number ??

Is there a closed form ???

Has this been studied before ??

Edit

Some remark or motivation :

Notice that when you naively try to analyse this , after you know it converges fast you get

Set all $$x_n = y$$ for large $$n$$ and some $$y > x > 1.$$

$$y = \frac{y^2 + y^2 + y^2}{y + y + y}$$

So

$$y = 3 y^2 / (3y)$$

$$y = y^2 / y$$

This is a tautology like $$z = z$$. Useless.

It seems all naive methods lead to such tautologies.

This is the one of the simplest possible with sum of squares ; rational function iterations of degree $$2.$$

This only informally suggests the dependance of the starting value $$x$$ ofcourse.

Also there seems no link to iterations that solves equations like Newton iterations.

Therefore the recursion fascinates me.

Also notice $$t(x)$$ is strictly increasing for $$x > 1.$$

• Is there some motivation behind this question? This looks like a "recurrence of cluster-type," which generalize Somos sequences. The caveat is that they must also yield integer values, whereas yours does not. – Alex R. Sep 25 '19 at 22:48
• Wolfram suggests: $-1 + e - 2 \sqrt{1 + e} + 6 \sqrt{1 + e^2} + \pi^2 + 5 \sqrt{1 + \pi}≈35.285088900886$ – Surb Sep 25 '19 at 22:54
• A nice one (among many other) : $\frac{867}{22 \log (\pi )}+\frac{3 \log (\pi )}{4}=35.2850889035603$. Can you provide more digits ? – Claude Leibovici Sep 26 '19 at 10:05
• $\frac{463986 e^e}{199273}=35.2850889000266$ is not bad – Claude Leibovici Sep 26 '19 at 10:11
• Based on computations in C++, I'm reasonably sure the value is $35.2850889285276615170428016199\dots$. – YiFan Oct 10 '19 at 10:11

A good approximate numerical value from mick : $$t(1)\simeq 35.28508892852..$$ and from my own computation : $$t(1)\simeq 35.28508892852\color{red}{76}.. \tag 1$$ I am not sure of the two last digits. So, my approximate value is the same as mick's value.
From Wolfram, reported by Surb : $$-1+e-2\sqrt{1+e}+6\sqrt{1+e^2}+\pi^2+5\sqrt{1+\pi}$$ $$\simeq 35.285088900886$$ From Claude Leibovici : $$\frac{867}{22\ln(\pi)}+\frac{3\ln(\pi)}{4}$$ $$\simeq 35.2850889035603$$ From Claude Leibovici : $$\frac{463986e^e}{199273}$$ $$\simeq 35.2850889000266$$ From my home-made ISC : $$-\frac{\left(\ln(5)-\sinh(e)\right)^2}{\sin\left(25/\ln(\gamma) \right)}\quad$$ , $$\gamma$$ is the Euler-Mascheroni constant. $$\simeq 35.285088928836$$ Even more accurately : $$-\frac{\left(\ln(5)-\sinh(e)\right)^2}{\sin\left(25/\ln(\gamma) \right)}-\frac{\gamma^3\pi}{\cosh\left(e^3/\sin(2) \right)}$$ $$\simeq 35.2850889285276$$ This is exactly the goal $$(1)$$ with 15 digits. Of course all this is not analytically true. An infinity of similar results can easily be obtained thanks to ISC as pointed out in the paper : https://fr.scribd.com/doc/14161596/Mathematiques-experimentales .