Lucas Numbers

Math 242 Modern Computational Mathematics

Generalizing the Fibonacci Numbers

The Lucas numbers (Subscript[L, 0]), (Subscript[L, 1]),(Subscript[L, 2]),... are defined similarly to the Fibonacci numbers, except that the first number is 2, not 0. Specifically:

(Subscript[L, 0])=2, (Subscript[L, 1])=1, and (Subscript[L, n])=(Subscript[L, n-2])+(Subscript[L, n-1]) for n>1.

Thus, the sequence of Lucas numbers is 2, 1, 3, 4, 7, 11, 18, 29, ....

We will modify our fibonacciList module so that it computes recursive sequences with any starting values that we specify. Specifically, our new code will be able to compute any generalized Fibonacci (or “gibonacci”) sequences defined as follows:

(Subscript[G, 0])=a, (Subscript[G, 1])=b, and (Subscript[G, n])=(Subscript[G, n-2])+(Subscript[G, n-1]) for n>1.

For example, if we set a=0 and b=1, then we get the Fibonacci sequence. If we set a=2 and b=1, then we get the Lucas sequence.

gibonacciList[a_,b_,n_]:=Module[{gibList=Table[0,n+1]},
(* set the initial values (Subscript[G, 0]) and (Subscript[G, 1]) *)

YOUR CODE HERE

(* compute (Subscript[L, 2]) through (Subscript[L, n]) and store them in the list *) 
Do[
gibList[[i]]=gibList[[i-1]]+gibList[[i-2]]
,{i,3,n+1}];

(* return the list *)
gibList
]

Now we can use this module to compute the Fibonacci numbers:

gibonacciList[0,1,10]

We can also compute the Lucas numbers 2,1,3,4,7,11,18,29,...

gibonacciList[2,1,10]

Today we will need both the Fibonacci and Lucas numbers. Let’s compute a big list of each and write two little function to fix the indexing to match the usual mathematical notation.

fibList=gibonacciList[0,1,10000];
f[n_]:=fibList[[n+1]]
lucList=gibonacciList[2,1,10000];
l[n_]:=lucList[[n+1]]

Print the first few numbers from each sequence to confirm that they are computed correctly:

f/@Range[0,10]
l/@Range[0,10]

Identities Involving Lucas Numbers

Explore the following question to discover identities involving the Lucas numbers.

1. Ratios of Lucas numbers

Examine the ratios ((Subscript[L, n+1])/(Subscript[L, n])) for various n. Do these ratios converge to some number as n increases? Formulate a conjecture supported by your computational evidence.

2. Fibonacci and Lucas

Compute (Subscript[F, n-1])+(Subscript[F, n+1]) for various n. What do you observe? Conjecture an identity. Can you prove your identity?

3. Lucas sums

Compute (∑_(k=0)^n)(Subscript[L, k]) for various integers n. What do you observe? Conjecture an identity.

4. More Fibonacci and Lucas

Compute the product (Subscript[F, n])(Subscript[L, n]) for various integers n. What do you observe? Conjecture an identity.

5. Lucas Squares

Compute (Subscript[L, n]^2) for various integers n. Compare with (Subscript[L, 2n]). What do you observe? Conjecture an identity.

6. Even More Fibonacci and Lucas

Compute (Subscript[L, m+1])(Subscript[F, n])+(Subscript[L, m])(Subscript[F, n-1]) for various integers m and n. What do you observe? Conjecture an identity.