Why the first $x$ decimal places of $(5+\sqrt{26})^{x}$ are following a pattern?

Question

$\sqrt{26}$ is irrational number, so the decimal places should show no pattern. But $(5+\sqrt{26})^{x}$ has these values:

x=1                    10.0990195135...
x=2                  101.99019513592...
x=3                1030.000970872871...
x=4              10401.9999038646405...
x=5            105050.00000951927653...
x=6          1060901.999999057405867...
x=7        10714070.0000000933352124...
x=8      108201601.99999999075799265...
x=9    1092730090.000000000915139071...
x=10 11035502501.9999999999093833742...
...

Where does the pattern of decimal places comes from?

Answer 1

Here is a big hint. Let $$f(x) = (5 + \sqrt{26})^x + (5 - \sqrt{26})^x.$$ Then $f(0) = 2$, $f(1) = 10$, and in general for $x = 2, 3, \ldots$, we have the recursion relation $$f(x)f(1) = f(x+1) + (5 + \sqrt{26})(5 - \sqrt{26})f(x-1) = f(x+1) - f(x-1),$$ or $$f(x+1) = 10f(x) + f(x-1).$$ Consequently, $f(x)$ is an integer for every nonnegative integer $x$.

Now consider the magnitude of $5 - \sqrt{26}$; in particular, note it is less than $1$. So successive integer powers of this number will tend toward $0$.

Answer 2

So take $a=5+\sqrt {26}, b=5-\sqrt {26}$ and note that $0\lt -b \lt 1$ (in fact $b \approx -0.1$)

Now note that $a+b=10$ and $ab=-1$ [whence $a=-\frac 1b$ and you can verify the estimate for $b$, because $a\gt 10$]

So $a$ and $b$ are roots of the quadratic $x^2-10x-1=0$ or $x^2=10x+1$ or $x^{n+2}=10x^{n+1}+x^n$

Let $Y_n=a^n+b^n$ then $Y_{n+2}=a^{n+2}+b^{n+2}=10a^{n+1}+a_n+10b^{n+1}+b_n=10Y_{n+1}+Y_n$

Or to summarise, we have the recurrence $Y_{n+2}=10Y_{n+1}+Y_n$

Now note that $Y_0=a^0+b^0=2; Y_1=a+b=10$ and you will find that the $Y_n$ are integers for all $n$ using the recurrence.

But also $Y_n=a^n+b^n$ so that $a^n=Y_n-b^n$ and $b$ is small and negative while $Y_n$ is an integer.

This shows why $a_n$ is nearly an integer and why it is alternately greater than or less than an integer.

The phenomena here come under the title "recurrence relations" or "difference equations" or the sums of powers of roots fo an equation. The simplest and best known is the Fibonacci sequence. This can all look like magic and "where does it come from?" but in fact is really quite elementary.

Answer 3

This phenomenon reminds of the Fibonacci sequence, that follows the recurrence

$$F_{n}=F_{n-1}+F_{n-2}$$ with initial values such that $F_n$ is always an integer.

$$0,1,2,3,5,8,13,21,34,\cdots$$

At the same time, it can be shown that

$$F_n=\frac{\phi^n-\psi^{-n}}{\sqrt 5}$$ where $\phi$ and $\psi$ are the roots of the characteristic question $x^2=x+1$. As $|\psi|<1$, we have that $|\psi^n|\ll1$, and $\dfrac{\phi^n}{\sqrt 5}$ is close to an integer.

As said by @heropup, your case corresponds to the recurrence

$$M_n=10M_{n-1}+M_{n-2}$$ and the whole sequence is $$-10,2,10,102,1030,10402,1050500,\cdots$$