DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-16

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The Fibonacci sequence $F_{0} = 0, F_{1} = 1$ and Lucas sequence $L_{0} = 2, L_{1} = 1$

Definition

The Fibonacci sequence $(F_{n})_{n \geq 0}$ and Lucas sequence $(L_{n})_{n \geq 0}$ are defined by

$F_{0} = 0, F_{1} = 1, L_{0} = 2, L_{1} = 1,$

and, for every $n \geq 0$ ,

$F_{n + 2} = F_{n + 1} + F_{n}, L_{n + 2} = L_{n + 1} + L_{n} .$

Both are order- $2$ recurrences in the convention of Constant-coefficient linear recurrences, their starting index and their characteristic polynomial, with characteristic polynomial $t^{2} - t - 1$ and reciprocal denominator $1 - x - x^{2}$ . The distinct initial pairs distinguish the two sequences.

Depends on

Constant-coefficient linear recurrences, their starting index and their characteristic polynomial

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 4 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Examples 4.1.2 and 4.7.16 (standard reference, not scraped)

The Fibonacci sequence F0=0,F1=1F0​=0,F1​=1 and Lucas sequence L0=2,L1=1L0​=2,L1​=1