Interesting or Fun Facts about Fibonacci Sequence

The Fibonacci sequence is a series where each number is the sum of the two preceding ones, starting from 0 and 1:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, .....

Fibonacci Numbers be derived using the formula: F_n = F_n-1 + F_n-2

Where,
• F_n​ represents the current Fibonacci number (the n-th term).
• F_n−1 represents the previous Fibonacci number (the (n−1)-th term).
• F_n−2​ represents the term before the previous Fibonacci number (the (n−2)-th term).
• Base condition: F₀ = 0 and F₁ = 1.

Below are some interesting facts about Fibonacci numbers.

Patterns in Fibonacci Numbers

• Pascal’s Triangle Connection: If you sum the diagonals of Pascal’s Triangle, you get the Fibonacci sequence. For example, the third diagonal sums to 1, 1, 2, 3, 5, and so on.

• The last digits of the Fibonacci Sequence repeat in a cycle of 60.

For example, the sequence of Fibonacci modulo 3 begins 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, ... The last two digits repeat with a period of 300, the last three with a period of 1500., last 4 digits with 15000, and the last 5 with 150,000. This is called Pisano Period.

• A Complete Sequence: Like Binary Representation where we represent every number as a sum of powers of 2, we can represent every every number as a sum of Fibonacci Numbers. For example: 19 = 13 + 5 + 119 = 13 + 5 + 119 = 13 + 5 + 1. This is known as a Zeckendorf Representation.

Factors and Divisibility Properties of Fibonacci Numbers

• Multiples of Fibonacci numbers: On careful observation, we can observe the following things:
  • Every 3-rd Fibonacci number is a multiple of 2
  • Every 4-th Fibonacci number is a multiple of 3
  • Every 5-th Fibonacci number is a multiple of 5
  • Every 6-th Fibonacci number is a multiple of 8
• Two consecutive Fibonacci Numbers never have any common factor.

• Every k-th Fibonacci number is a multiple of F_k
  This point can also be written as F_nk is a multiple of F_k for all values of n and k greater than 1.

Note: Every number is a factor of some Fibonacci number and there are infinitely many Fibonacci numbers with any given factor.

Fibonacci Numbers with Index Divisibility

Certain Fibonacci numbers are divisible by their corresponding index numbers. For example:

• F(1) = 1, which is divisible by 1,
• F(5) = 5, which is divisible by 5,
• F(12) = 144, which is divisible by 12,
• F(24) = 46368, which is divisible by 24,
• F(25) = 75025, which is divisible by 25.

This type of index number follows a certain pattern:

1, 5, 12, 24, 25, 36, 48, 60, 72, 84, 96, 108, 120, 125, 132, ..... 

This series has all powers of 5 and all multiples of 12.

Mathematical Properties of Fibonacci Numbers

• GCD and Fibonacci: GCD of m-th and n-th Fibonacci Numbers is equal to GCD of m and n. In other words GCD(f(m), f(n)) = (GCD(m, n)). The above property holds because Fibonacci Numbers follow the Divisibility Sequence, i.e., if m divides n, then f(m) also divides f(n). For example, f(3) = 2 and every third third Fibonacci Number is even.

• Matrix Representation: There's a special matrix (transformation matrix) that represents how Fibonacci numbers work.
  It looks like this: \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}This matrix captures the Fibonacci relationship. If we multiply this matrix by itself multiple times, it can give us Fibonacci numbers.
  To find the Nth Fibonacci number we need to multiply the transformation matrix (n-1) times, the matrix equation for the Fibonacci sequence looks like this:
  \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{n-1} = \begin{pmatrix} F(n) & F(n-1) \\ F(n-1) & F(n-2) \end{pmatrix} After raising the transformation matrix to the power n - 1, the top-left element F(n) will give the nth Fibonacci number.

• Addition Rule: The (m + n)-th Fibonacci Number can be directly obtained using F(m + n) = F(m) x F(n + 1) + F(m - 1) x F(n). This formula can be directly derived from the matrix representation discussed in the previous point.

• Fibonacci Primes: While most Fibonacci numbers are composite, some of them are prime numbers. These are known as Fibonacci primes.
  • For example, 2, 3, 5, 13, 89 are Fibonacci primes.

• Lucas Numbers: A similar sequence, known as Lucas Numbers, starts with 2 and 1 instead of 0 and 1, but they share many properties with the Fibonacci sequence. Both sequences satisfy the same recurrence relation.

• The difference between the product of the previous and next and the square of n-th Fibonacci is (-1)ⁿ. In other words, the value of f(n - 1)*f(n + 1) - f(n)*f(n) is (-1)ⁿ. Please refer to Cassini’s Identity for details.

• Carmichael's Theorem: For n greater than 12, the nth Fibonacci number F(n) has at least one prime divisor that does not divide any earlier Fibonacci number.

Connection to the Goldenintegers Ratio

• Golden Ratio Connection: The ratio of consecutive Fibonacci numbers approaches the Golden Ratio (approximately 1.618) as for large integers. We can use this fact to find the nth Fibonacci number (for a large n) as the golden ratio
• raised to the power n.

• Binet's Formula: There is a direct formula to find the n-th Fibonacci Number: F_n = \frac{1}{\sqrt{5}}((\frac{1 + \sqrt{5}}{2})^n -(\frac{1 - \sqrt{5}}{2})^n)
  Here,\varphi = \frac{1+\sqrt{5}}{2} is called Golden Ratio. And for large values of n, this formula can be easily approximated to the golden ratio raised to the power n.

Summation Properties of Fibonacci Numbers

• Sum of Fibonacci Numbers: The sum of the first n Fibonacci numbers is 2F_(n+1) - 1. In other words, F(0) + F(1) + F(2 ) +⋯+ F(n) = F(n + 2) - 1.
  • For example, the sum of the first 5 Fibonacci numbers (0, 1, 1, 2, 3) is 7, which is F(7) - 1 = 8 - 1.
• Sum of Even Positioned Fibonacci Numbers: The sum of the first n even positioned Fibonacci numbers is F_(2n+1) - 1. In other words F(2) + F(4) + F(6) + ⋯ + F(2n) = F(2n+1)
  • For example, summing the even Fibonacci numbers 1, 3, and 8 gives 12, and F(7) - 1 = 13 - 1.

Fibonacci in Nature and Art

• Fibonacci in Nature: Fibonacci numbers appear in biological settings, such as the arrangement of leaves around a stem, the branching of trees, the arrangement of a pine cone's scales, and even in the spiral patterns of shells and galaxies.

• Fibonacci Numbers and Music: The Fibonacci sequence can be found in the structure of musical compositions, including the number of notes, the timings of phrases, and even the organization of musical scales.

• Fibonacci in Art: The Fibonacci sequence is closely linked to the concept of aesthetic proportions and is often found in art and architecture, such as the Parthenon and the works of artists like Leonardo da Vinci.

• Nature's Efficiency: The Fibonacci sequence provides an efficient way for plants to maximize sunlight exposure and seed arrangement in flowers. The spiral patterns that occur follow Fibonacci numbers to optimize packing.

Related Reads:

• Real Life Applications of Fibonacci Series.
• Interesting Facts about Pascals Triangle.
• Interesting Facts about Prime Numbers

Interesting Problems Based on Fibonacci Numbers

1. Stair Climbing: There are n stairs, and a person standing at the bottom wants to climb stairs to reach the top. The person can climb either 1 stair or 2 stairs at a time, the task is to count the number of ways that a person can reach at the top.
2. Binary Strings Without Consecutive 1s: Given a positive integer N, count all possible distinct binary strings of length N such that there are no consecutive 1’s.
3. Dudeney's Cow Problem: If a cow has her first female calf at age two, and then produces another female calf each year after that, how many female calves will there be after 12 years, assuming no deaths?