The sequence of Lucas numbers is {1, 3, 4, 7, 11, 18, 29, 47, ...}. This sequence arises from the following recurrence relation: L(1)=1, L(2)=3, L(n+2)=L(n+1)+L(n). Compare this to the Fibonacci sequence, whose recurrence relation is: F(1)=1, F(2)=1, F(n+2)=F(n+1)+F(n).

Some interrelated properties of the Lucas and Fibonacci numbers

  • F(2n)=F(n)L(n)
  • L(2n)=L^2(n)-2(-1)^n
  • L(0)+L(1)+...+L(n)=L(n+2)-1
  • L(n)=F(n-1)+F(n+1)
  • 2F(m+n)=F(m)L(n)+F(n)L(m)
  • 5F(n)=L(n-1)+L(n+1)
  • 2L(m+n)=L(m)L(n)+5F(m)F(n)

And the list continues. The ratio of one Lucas number to the next tends to the golden ratio, just like the ratio of one Fibonacci number to the next. The ratio of L(n)/F(n) tends to sqrt(5) (the square root of 5) as n approaches infinity.

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