numbers (version 0.3-3)

fibonacci: Fibonacci and Lucas Series

Description

Generates single Fibonacci numbers or a Fibonacci sequence; generates a Lucas series based on the Fibonacci series.

Usage

fibonacci(n, sequence = TRUE)
lucas(n)

Arguments

n

integer.

sequence

logical; default: True.

Value

• A single integer, or a vector of integers.

Details

Generates the n-th Fibonacci number, or the whole Fibonacci sequence from the first to the n-th number. Generates only the sequence of Lucas numbers.

The recursive version (sequence=FALSE) is extremely slow for values n>=30. To get the n-th Fibonacci number for larger n values, use fibonacci(n)[n].

Examples

fibonacci(0)                            # 1
fibonacci(2)                            # 2
fibonacci(2, sequence = TRUE)           # 1 2

lucas(0)                                # 2
lucas(2)                                # 1 3

# Golden ratio
F <- fibonacci(25, sequence = TRUE)     # ... 75025 121393
f25 <- F[25]/F[24]                      #     1.618033989
phi <- (sqrt(5) + 1)/2
abs(f25 - phi)                          # 7.945178e-11

# Fibonacci numbers w/o iteration
  fibo <- function(n) {
    phi <- (sqrt(5) + 1)/2
    fib <- (phi^(n+1) - (1-phi)^(n+1)) / (2*phi - 1)
    round(fib)
  }
fibo(30:33)                             # 1346269 2178309 3524578 5702887

  luca <- function(n) {
    phi <- (sqrt(5) + 1)/2
    floor(phi^n + 1/2)
  }
luca(30:33)                             # 1860498 3010349 4870847 7881196

# Compare recursive with iterative approach:
# system.time(F30 <- fibonacci(30, sequence = FALSE)      # user: 17.075 s
# system.time(F30 <- fibonacci(30, sequence = TRUE)[30])  # user:  0.006 s