LucasL

The nth Lucas number: the Fibonacci-style recurrence started from $L_0 = 2$, $L_1 = 1$, closely related to Fibonacci.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaLucas number↗MathWorldLucasNumber↗WikidataQ2503280↗OEISA000032↗Wolfram LanguageLucasL✓↗

LucasL(n)the nth Lucas number $L_n$.

SageMathlucas_number2(n, 1, -1)✓↗SymPylucas(n)↗JuliaCombinatorics.lucasnum(n)

LucasL(nu)a real (non-integer) index via Binet's formula, $L_\nu = \varphi^\nu + \cos(\pi\nu)\varphi^{-\nu}$.

SageMathlucas_number2(nu, 1, -1)✓↗SymPylucas(nu)↗JuliaCombinatorics.lucasnum(nu)

LucasL(n, x)the Lucas polynomial $L_n(x)$, from $L_n(x) = xL_{n-1}(x) + L_{n-2}(x)$ with $L_0(x) = 2$, $L_1(x) = x$, exact for a nonnegative integer n.

Typed signatures

Type	Package	Applies
(integer) -> integer	compute-engine	always
(number, any?) -> any	number-theoryreplaces compute-engine	always
(value) -> value	adelesreplaces number-theory	on ProfiniteNumber

Domain: Sequences

Details

• Defined by the same recurrence as Fibonacci, $L_n = L_{n-1} + L_{n-2}$, but started from $L_0 = 2$, $L_1 = 1$.
• Closed form: $L_n = \varphi^n + \psi^n$, where $\varphi = \frac{1+\sqrt5}{2}$ and $\psi = \frac{1-\sqrt5}{2}$.
• Related to Fibonacci by $L_n = F_{n-1} + F_{n+1}$ and, conversely, $F_n = \dfrac{L_{n-1} + L_{n+1}}{5}$.
• Consecutive ratios $L_{n+1}/L_n$ converge to the golden ratio, just as they do for Fibonacci.
• Extends to negative n via $L_{-n} = (-1)^n L_n$.
• A list of indices is threaded over element-wise; a real (non-integer) index evaluates numerically via Binet's formula, and a symbolic second argument gives the Lucas polynomial.

Examples

Basic #

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In$\mathrm{LucasL}(0)$Out$2$

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In$\mathrm{LucasL}(1)$Out$1$

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In$\mathrm{LucasL}(5)$Out$11$

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In$\mathrm{LucasL}(10)$Out$123$

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In$\operatorname{Series}(\mathrm{LucasL}(\frac{1}{2}, x), x, 0, 5)$Out$\mathrm{LucasL}(\frac{1}{2}, 0)+x\mathrm{LucasL}^{(0, 1)}(\frac{1}{2}, 0)+\frac{1}{2}\mathrm{LucasL}^{(0, 2)}(\frac{1}{2}, 0)x^2+\frac{1}{6}\mathrm{LucasL}^{(0, 3)}(\frac{1}{2}, 0)x^3+\frac{1}{24}\mathrm{LucasL}^{(0, 4)}(\frac{1}{2}, 0)x^4+\frac{1}{120}\mathrm{LucasL}^{(0, 5)}(\frac{1}{2}, 0)x^5+O\left(x^6\right)$

Scope #

Exact at any size

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In$\mathrm{LucasL}(100)$Out$792\,070\,839\,848\,372\,253\,127$

An odd negative index flips the sign: $L_{-11} = -L_{11}$

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In$\mathrm{LucasL}(-11)$Out$-199$

A real index uses $L_\nu = \varphi^\nu + \cos(\pi\nu)\,\varphi^{-\nu}$

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In$\mathrm{LucasL}(2.333\,333\,333\,333\,333\,5)$Out$3.236\,211\,879\,491\,621\,3$

The two-argument Lucas polynomial $L_7(x)$, from $L_n(x) = x L_{n-1}(x) + L_{n-2}(x)$

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In$\mathrm{LucasL}(7, x)$Out$x^7+7x^5+14x^3+7x$

The Lucas polynomial at $x = 1$ is the Lucas number: $L_{143}(1) = L_{143}$

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In$\mathrm{LucasL}(143, 1)$Out$767\,772\,505\,664\,398\,093\,937\,756\,525\,279$

Special value of the Lucas polynomial: $L_1(0) = 0$

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In$\mathrm{LucasL}(1, 0)$Out$0$

Special value of the Lucas polynomial: $L_0(0) = 2$

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In$\mathrm{LucasL}(0, 0)$Out$2$

$L_2(x) = x^2 + 2$ threads element-wise over a matrix argument

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In$\mathrm{LucasL}(2, \bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{9}{4}, 3\bigr\rbrack, \bigl\lbrack2, \frac{9}{4}\bigr\rbrack\bigr\rbrack$

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\mathrm{LucasL}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\bigl\lbrack1, 3, 4, 7\bigr\rbrack$

Lucas numbers extend to rational index; 50-digit evaluation

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In$N(\mathrm{LucasL}(\frac{11}{3}), 50)$Out$5.923\,962\,652\,961\,955\,410\,135\,697\,862\,194\,012\,628\,751\,985\,542\,236\,2$

Complex index gives a complex value

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In$N(\mathrm{LucasL}(1+\frac{1}{2}i))$Out$0.065\,338\,425\,470\,823\,424\,320\,359\,862\,424\,055\,077\,050\,251\,860\,953\,103\,4+0.755\,094\,920\,107\,233\,972\,742\,981\,419\,715\,215\,424\,101\,179\,285\,156\,060\,2i$

The Lucas polynomial of odd index is an odd function of $x$

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In$\mathrm{LucasL}(1, -x)=-\mathrm{LucasL}(1, x)$Out$\mathrm{True}$

Even index gives an even polynomial in $x$

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In$\mathrm{LucasL}(2, -x)=\mathrm{LucasL}(2, x)$Out$\mathrm{True}$

Threads over a list of indices

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In$\mathrm{LucasL}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack)$Out$\bigl\lbrack1, 3, 4, 7, 11\bigr\rbrack$

Generating function $\frac{2-x}{1-x-x^2}$

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In$\mathrm{GeneratingFunction}(\mathrm{LucasL}(n), n, x)$Out$\frac{2-x}{-x^2-x+1}$

Real coefficients, so it commutes with complex conjugation

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In$\mathrm{LucasL}(1, \overline{z})=\overline{\mathrm{LucasL}(1, z)}$Out$\mathrm{True}$

Applied to a matrix as a function, here $x^2+2$ on an upper triangular matrix

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In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{LucasL}(2, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1/2, -1\bigr\rbrack, \bigl\lbrack0, 1/2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack\frac{9}{4}, -1\bigr\rbrack, \bigl\lbrack0, \frac{9}{4}\bigr\rbrack\bigr\rbrack$

Applications #

Consecutive ratios $L_{n+1}/L_n$ converge to the golden ratio too. See Fibonacci

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In$\frac{\mathrm{LucasL}(15)}{\mathrm{LucasL}(14)}$Out$\frac{1\,364}{843}$

First index whose Lucas number exceeds $10^6$

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In$\mathrm{NestWhile}(\operatorname{\_1}↦\operatorname{\_1}+1, 1, \operatorname{\_1}↦\mathrm{LucasL}(\operatorname{\_1})≤10^6)$Out$29$

Properties #

$L_n^2 - 5F_n^2 = 4(-1)^n$, here at $n = 10$. See Fibonacci

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In$\mathrm{LucasL}(10)^2-5\mathrm{Fibonacci}(10)^2$Out$4$

$L_n = F_{n-1} + F_{n+1}$, here at $n = 6$

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In$\mathrm{LucasL}(6)=\mathrm{Fibonacci}(5)+\mathrm{Fibonacci}(7)$Out$\mathrm{True}$

Conversely $F_n = \dfrac{L_{n-1} + L_{n+1}}{5}$

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In$\mathrm{Fibonacci}(6)=\frac{\mathrm{LucasL}(5)+\mathrm{LucasL}(7)}{5}$Out$\mathrm{True}$

A Cassini-style identity $L_{n-1}L_{n+1} - L_n^2 = 5(-1)^{n+1}$, here at $n = 5$

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In$\mathrm{LucasL}(4)\mathrm{LucasL}(6)-\mathrm{LucasL}(5)^2$Out$5$

Negative indices are supported via $L_{-n} = (-1)^n L_n$

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In$\mathrm{LucasL}(-1)$Out$-1$

$L_2=1+2$

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In$\mathrm{LucasL}(2)$Out$3$

Known value · Wolfram MathematicalFunctionData, LucasL

Each term is the sum of the previous two

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In$\mathrm{LucasL}(3)$Out$4$

Known value · Wolfram MathematicalFunctionData, LucasL

$L_4=3+4$

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In$\mathrm{LucasL}(4)$Out$7$

Known value · Wolfram MathematicalFunctionData, LucasL

$L_6=7+11$

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In$\mathrm{LucasL}(6)$Out$18$

Known value · Wolfram MathematicalFunctionData, LucasL

$L_7=11+18$

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In$\mathrm{LucasL}(7)$Out$29$

Known value · Wolfram MathematicalFunctionData, LucasL

Lucas numbers grow by about the golden ratio each step

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In$\mathrm{LucasL}(8)$Out$47$

Known value · Wolfram MathematicalFunctionData, LucasL

$L_9=29+47$

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In$\mathrm{LucasL}(9)$Out$76$

Known value · Wolfram MathematicalFunctionData, LucasL

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In$\sum_{k=1}^2\frac{(-1)^{k}}{\mathrm{LucasL}(k)\mathrm{LucasL}(k+2)}$Out$-\frac{17}{84}$

Known value · Wolfram MathematicalFunctionData, LucasL, FunctionalEquations

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In$\mathrm{LucasL}(2)^2-\mathrm{LucasL}(1)\mathrm{LucasL}(3)$Out$5$

Known value · Wolfram MathematicalFunctionData, LucasL, FunctionalEquations

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In$\sum_{k=1}^3\frac{(-1)^{k}}{\mathrm{LucasL}(k)\mathrm{LucasL}(k+2)}$Out$-\frac{52}{231}$

Known value · Wolfram MathematicalFunctionData, LucasL, FunctionalEquations

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In$N(\mathrm{LucasL}(1/2)^2-\mathrm{LucasL}(-\frac{1}{2})\mathrm{LucasL}(\frac{3}{2}))$Out$-3.937\,936\,909\,913\,409\,042\,4\cdot10^{-21}$

Known value · Wolfram MathematicalFunctionData, LucasL, FunctionalEquations

Neat examples #

$L_{30}$ already tops 1.8 million, growing at the same rate as Fibonacci

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In$\mathrm{LucasL}(30)$Out$1\,860\,498$

About

Examples	41
Checked against	Wolfram Language 41 agreeSageMath 23 agree6 errorSymPy 25 agree1 differ3 unevaluatedmpmath 27 agreeJulia 21 agree6 error
Crosswalk	WikipediaMathWorldWikidataOEISWolfram LanguageSageMathSymPyJulia

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

F(n − 1) + F(n + 1): Lucas' number for integer n and its continuation otherwise.

See also: Fibonacci