Built-in Mathematica Symbol

LucasL

LucasL[n]
gives the Lucas number

.

LucasL[n, x]
gives the Lucas polynomial

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

The satisfy the recurrence relation with , .

For any complex value of n the are given by the general formula , where is the golden ratio.

The Lucas polynomial is the coefficient of in the expansion of .

The Lucas polynomials satisfy the recurrence relation .

LucasL can be evaluated to arbitrary numerical precision.

LucasL automatically threads over lists.

EXAMPLES

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Basic Examples (1)

Compute Lucas numbers:

In[1]:=

Out[1]=

Scope (7)

Evaluate large Lucas numbers:

Lucas numbers of negative arguments:

Non-integer arguments:

Complex arguments:

LucasL threads element-wise over lists:

Series expansion at a generic point:

TraditionalForm formatting:

Generalizations & Extensions (1)

Lucas polynomials:

Applications (5)

Solve the Fibonacci recurrence equation:

Find ratios of successive Lucas numbers:

Compare with continued fractions:

Convergence to the Golden Ratio:

Calculate the number of ways to write an integer as a sum of Lucas numbers

:

Plot the counts for the first hundred integers:

Find the first Lucas number above 1000000:

First few Lucas pseudoprimes:

Properties & Relations (5)

Expand in terms of elementary functions:

Limiting ratio:

Explicit recursive definition:

Simplify some expressions involving Lucas numbers:

Generating function:

Extract Lucas numbers as coefficients:

Possible Issues (2)

Large arguments can give results too large to be computed explicitly:

Results for integer arguments may not hold for non-integers:

Neat Examples (2)