<s>[INST] General solution expressed in $a_0$ and $a_1$ of a Fibonacci-like sequence? 
What is the general solution expressed in $a_0$ and $a_1$ of a Fibonacci-like sequence ?

I mean if $a_0,a_1$ are given and $a_{n+1}:=a_n+a_{n-1}$
$(\begin{array}{cc}a_n&a_{n-1}\end{array})=(\begin{array}{cc}a_{n-1}&a_{n-2}\end{array})\left(\begin{array}{cc}1&1\\1&0\end{array}\right)=(\begin{array}{cc}a_{1}&a_{0}\end{array})\left(\begin{array}{cc}1&1\\1&0\end{array}\right)^{n-1}$ 
or according to this formula; $a_n=a_1\sum\limits_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\binom{n-k-1}{k}+a_0\sum\limits_{k=0}^{\left\lfloor\frac{n-2}{2}\right\rfloor}\binom{n-k-2}{k}$
 [/INST] The general solution will be as follows:
$a_n = \dfrac{\varphi_2a_0 - a_1}{\varphi_2 - \varphi_1}\varphi_1^{n} + \dfrac{a_1 - \varphi_1a_0}{\varphi_2 - \varphi_1}\varphi_2^{n}$, where $\varphi_{1,2} = \dfrac{1 \pm \sqrt{5}}{2}$ -- roots of equation $\varphi^2-\varphi-1 = 0$.  You can check it by substituting this formula to the recurrence.
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