Fibonacci sequence questions

Question

I want to show that "Out of the first 450 Fibonacci numbers, the odd number is twice as many as even number." with Mathematica.

Can you solve it please?

Answer 1

First the Fibonacci numbers

out = Table[Fibonacci[n], {n, 450}];

then counting....

Mod[#, 2] & /@ out // Counts

(<|1 -> 300, 0 -> 150|>)

Answer 2

Let's find the conditions under which a Fibonacci number is odd or even. Happily, Reduce has us covered:

isOdd = Reduce[Mod[Fibonacci[n], 2] == 1, n, Integers]
(* Element[C[1], Integers] && (n == 1 + 3 C[1] || n == 2 + 3 C[1]) *)

isEven = Reduce[Mod[Fibonacci[n], 2] == 0, n, Integers]
(* Element[C[1], Integers] && (k == 0 && n == 3*C[1]) *)

This says for all integers $ n $, $ F_n \equiv 0\,\left(\bmod 2\right) $ when $ n \equiv 0 \left(\bmod 3\right) $.

Now, let's see how many instances there are of each for $n$ between $1$ and $450$.

evens = Length@FindInstance[isEven && 1 <= n <= 450, {n, C[1]}, Integers, 450]
(* 150 *)

odds = Length@FindInstance[isOdd && 1 <= n <= 450, {n, C[1]}, Integers, 450]
(* 300 *)

Now that we've come this far, Mathematica can also help us with the basic arithmetic:

2 * evens == odds
(* True *)

Answer 3

Similar to mgamer's nice answer:

Mod[Array[Fibonacci, 450], 2]//Counts

<|1 -> 300, 0 -> 150|>

Edit

To 'borrow' the neat suggestion made by rcollyer in a comment to rgamer's answer:

Array[Fibonacci, 450] // CountsBy[OddQ]

<|True -> 300, False -> 150|>

Edit 2

lhf points out in a comment that "$F_n$ is even iff $n$ is a multiple of 3"

Table[Fibonacci[n], {n,3, 450,3}] // CountsBy[EvenQ] 

<|True -> 150|>

A very interesting one, it seems to me. As long as $n$ is a multiple of 3, then "the odd number is twice as many as even number" as the OP asks for $n$ equal to 450.

Answer 4

Array[Fibonacci, 450, 1, CountsBy[OddQ] @* List]

<|True -> 300, False -> 150|>

Also

Array[Fibonacci /* OddQ, 450, 1, List /* Counts]

<|True -> 300, False -> 150|>

and

450 // Range /* Fibonacci /* CountsBy[OddQ]

<|True -> 300, False -> 150|>