Lucas number
From Free net encyclopedia
The Lucas numbers are a integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Much like the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate predecessors. Consequently, the ratio between two consecutive Lucas numbers converges to the golden ratio.
However, the first two Lucas numbers are L0 = 2 and L1 = 1 instead of 0 and 1, and the properties of the Lucas sequence are therefore somewhat different from those of the Fibonacci sequence. The sequence of Lucas numbers begins:
The Lucas numbers are related to the Fibonacci numbers by the identities
- <math>L_n = F_{n-1}+F_{n+1},</math>
- <math>F_{2n} = L_n F_n.</math>
Their closed formula is given as:
- <math>L_n = \varphi^n + (-\varphi)^{-n}</math>
where <math>\varphi</math> is the golden ratio.