DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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