Lucas number: another beautiful patterns of sequence


Formula for obtaining the Lucas numbers


The Lucas numbers are numbers that form Fibonacci integer sequence, each Lucas number is defined to be the sum of its immediate previous term. The Lucas number also known as Lucas series was named after the mathematician François Edouard Anatole Lucas. He came about the Lucas number by studying the Fibonacci numbers and establishing the fact that Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values.
Some of the first few Lucas sequence are given by 2,1,3,4,7,11,18,29,47,76,123,....
The first two Lucas sequence are L0=2 and L1=1. The Lucas numbers is defined by Ln=Ln−1+Ln−2 for n>1,n∈N.
Therefore, 
  1. L2=L2−1+L2−2=L1+L0=1+2=3
  2. L3=L3−1+L3−2=L2+L1=3+1=4
  3. L4=L4−1+L4−2=L3+L2=4+3=7
The remaining numbers are 11,18,29,47,76,123,199,... and are also generated same way the above ones are generated. 
We can extend the Lucas number to negative integers by using the definition Ln−2=Ln−Ln−1, this gives a doubly infinite sequence ...,−11,7,−4,3,−1,2,1,3,4,7,11,... for −5≤n≤5. The formula for terms with negative indices is given by Ln=(−1)nLn. 
The Lucas numbers are related to the Fibonacci numbers by many identities for example:
  1.  Ln=Fn−1+Fn+1=Fn+2Fn−1=Fn+2−Fn−2
  2.  Lm+n=Lm+1Fn+LmFn−1
  3.  L2n=5F2n+4(−1)n approaches +∞, the ratio LnFn approaches √5.
  4.  F2n=LnFn
  5.  L2n=5F2n+2(−1)n=L2n−2(−1)n
  6.  Fn+k+(−1)kFn−k=LkFk
  7.  Ln+k−(−1)kLn−k=5FnFk in particular Fn=Ln−1+Ln+15.
Their closed formula is given by
Ln=φn+(1−φ)n=φn+(−φ)−1
=(1+√52)n+(1−√52)nwhere\varphi$ is defined as the golden ratio. 
A formula for φn is thus obtained as 
φn=Ln+Fn√52.