Negafibonacci Numbers Generator

The Negafibonacci numbers represent the extension of the classic Fibonacci sequence to negative integers. While standard Fibonacci numbers are computed by moving forward ($n \ge 0$), the Negafibonacci sequence moves backward into negative indices ($n < 0$), utilizing a mathematically rigorous signed relation that preserves the additive properties of the Fibonacci recurrence.

Mathematical Definition

The Negafibonacci sequence is defined recursively by rewriting the classic Fibonacci recurrence relation:

By setting $n = 1$, we can find $F_{-1}$:
$F_{-1} = F_1 - F_0 = 1 - 0 = 1$
By setting $n = 0$, we find $F_{-2}$:
$F_{-2} = F_0 - F_{-1} = 0 - 1 = -1$

This backward calculation yields a beautiful sequence with alternating signs:
0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, ...

The Parity and Sign Formula

There is a highly elegant closed-form relationship between negative index terms and positive index terms:

This formula reveals a beautiful parity behavior:

• For **odd negative indices** (e.g. $n = 1, 3, 5$), the sign is positive: $F_{-n} = F_n$.
• For **even negative indices** (e.g. $n = 2, 4, 6$), the sign is negative: $F_{-n} = -F_n$.

Frequently Asked Questions (FAQs)