This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# LucasL

 LucasL[n] gives the Lucas number . LucasLgives the Lucas polynomial .
• 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.
Compute Lucas numbers:
Compute Lucas numbers:
 Out[1]=
 Scope   (7)
Evaluate large Lucas numbers:
Lucas numbers of negative arguments:
Non-integer arguments:
Complex arguments:
Series expansion at a generic point:
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:
Expand in terms of elementary functions:
Limiting ratio:
Explicit recursive definition:
Simplify some expressions involving Lucas numbers:
Generating function:
Extract Lucas numbers as coefficients:
Large arguments can give results too large to be computed explicitly:
Results for integer arguments may not hold for non-integers: