Why Fibonacci sequence start at $0$, Tribonacci sequence with $0,0$, Tetranacci with $0,0,0$, etc. [ref OEIS] Has any good reasons for that?
These sequences arise in generalization of Pascal Triangle as diagonal sums and there they start at $1$.
Pascal triangle:
$$\begin{array}{} \color{red}1& \color{blue}0& \color{green}0& \color{cyan}0&\dots\\ \color{blue}1& \color{green}1& \color{cyan}0& \color{magenta}0&\dots\\ \color{green}1& \color{cyan}2& \color{magenta}1& 0&\dots\\ \color{cyan}1& \color{magenta}3& 3& \color{red}1&\dots\\ \vdots&\vdots&\vdots&\vdots&\ddots \end{array}$$
diagonal sum gives $\color{red}1,\color{blue}1,\color{green}2,\color{cyan}3,\color{magenta}5,8,\color{red}{13}\dots$ Fibonacci sequence
First generalization:
$$\begin{array}{r} \color{red}1& \color{blue}0& \color{green}0& \color{cyan}0& \color{magenta}0& 0&\color{red}0&\dots\\ \color{blue}1& \color{green}1& \color{cyan}1& \color{magenta}0& 0& \color{red}0&\color{blue}0&\dots\\ \color{green}1& \color{cyan}2& \color{magenta}3& 2& \color{red}1& \color{blue}0&\color{green}0&\dots\\ \color{cyan}1& \color{magenta}3& 6& \color{red}7& \color{blue}6& \color{green}3&\color{cyan}1&\dots\\ \color{magenta}1&4&\color{red}{10}&\color{blue}{16}&\color{green}{19}&\color{cyan}{16}&\color{magenta}{10}&\dots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots \end{array}$$
gives sequence of diagonal sum $\color{red}1,\color{blue}1,\color{green}2,\color{cyan}4,\color{magenta}7,\dots$, Tribonacci sequence, etc.