Fibonacci series and Lucas series

The Fibonacci Series and the Lucas Series
The following table shows terms 0 to 20 of the Fibonacci series. With each nth term is shown the 2nth term. As the table shows, any Fibonacci term F2n is a multiple of Fn.

The right hand column gives the result of dividing F2n by Fn. This produces the series (starting at n = 1), 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, ... .

This series has the same property as the Fibonacci series in that each term is the sum of the two before it. It is called the Lucas series, after Édouard Lucas (1842-1891), a French mathematician who extensively studied both series. He also invented the Tower of Hanoi puzzle. The figures in the T(2n) / T(n) column show a number series with the same property as the Fibonacci: each number in the series is the sum of the two before it. However, instead of beginning 0, 1,… it begins 2, 1,…, with 2 instead of 0 as the 0th term. This series is called the Lucas series. Terms 0 to 40 of both series are shown in the table below.