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gives the Lucas number .
gives 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:
Click for copyable input
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:
Lucas polynomials:
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:
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