Fibonacci seed at index one

Read the equation

The second seed is F₁=1. Combined with F₀=0, it yields F₂=1, F₃=2, F₄=3, and so on under the recurrence. Different initial seeds would define a different sequence, even if a similar ratio limit sometimes holds.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

with conjugate $\golden^{-1} = \golden - 1 \approx 0.618033988749895$.
\end{definition}

\begin{definition}[Fibonacci Sequence]
$\fib{0} = 0$, $\fib{1} = 1$, and $\fib{n} = \fib{n-1} + \fib{n-2}$ for $n \geq 2$.
\end{definition}

The ratio of consecutive Fibonacci numbers converges to $\golden$:
\begin{equation}
\lim_{n \to \infty} \frac{\fib{n}}{\fib{n-1}} = \golden
\end{equation}

\fib{1} = 1

Related builds