Definition:Fibonacci Number/Negative

Definition

The definition of Fibonacci numbers for negative integers is an extension of the definition for positive integers:

$F_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n + 2} - F_{n + 1} & : n < 0 \end{cases}$

for all $n \in \Z$.

Sequence

The sequence of Fibonacci numbers for negative index begins:

$\ldots, 89, -55, 34, -21, 13, -8, 5, -3, 2, -1, 1, 0$

Sources

• 1957: George Bergman: Number System with an Irrational Base (Math. Mag. Vol. 31, no. 2: pp. 98 – 110)  www.jstor.org/stable/3029218
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $8$

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