Finding the LFSR and connection polynomial for binary sequence. I have written a C implementation of the Berlekamp-Massey algorithm to work on finite fields of size any prime. It works on most input, except for the following binary GF(2) sequence:

$0110010101101$ producing LFSR $\langle{}7, 1 + x^3 + x^4 + x^6\rangle{}$

i.e. coefficients $c_1 = 0, c_2 = 0, c_3 = 1, c_4 = 1, c_5 = 0, c_6 = 1, c_7 = 0$ 

However, when using the recurrence relation

\begin{equation}
s_j = (c_1s_{j-1} + c_2s_{j-2} + \cdots + c_Ls_{j-L}) \mbox{ for } j \geq L.
\end{equation}
to check the result, I get back:

0110010001111, which is obviously not right.

Using the Berlekamp-Massey Algorithm calculator they say the (I believe) characteristic polynomial should be $x^7 + x^4 + x^3 + x^1$. Which, according to my paper working, the reciprocal should indeed be $1 + x^3 + x^4 + x^6$.

What am I doing wrong?  Where is my understanding lacking?
This is probably just a difference in notation than any failure in
understanding or implementation. Some people define recurrence relations with subscripts in reverse order than others; the original description given by Berlekamp in his 1968 book Algebraic Coding
Theory began counting from $1$ instead of $0$ etc.  Observe that
$$x^7 + x^4 + x^3 + x^1 = x^7(1 + x^{-3} + x^{-4} + x^{-6})$$
in comparison to your $1 + x^3 + x^4 + x^6$ which you say is the
correct reciprocal of what the web site's answer should be.
So I would say that the web site seems to be following
Berlekamp's original description and giving you an answer that is "off-by-one".