"In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones."
A sequence of numbers where each term is the sum of the two preceding terms.
Basic Arithmetic and Algebra: This is the foundational knowledge required to understand the basics of the Fibonacci sequence, including concepts such as addition, subtraction, multiplication, division, and the use of variables.
Number Sets: Understanding the different types of number sets, such as natural numbers, integers, rational numbers, and real numbers, is essential for understanding the properties and patterns of the Fibonacci sequence.
Properties of Sequences: Understanding the properties of sequences, such as convergence and divergence, monotonicity, and boundedness, will help you understand how the Fibonacci sequence behaves.
Geometric Sequences: Study of geometric sequence, its definition, formula, and properties.
Recurrence Relations: Recurrence relation is in a way the backbone of the Fibonacci sequence. Understanding recurrence relations and how they are used in generating the Fibonacci sequence is crucial to understand the sequence itself.
Limits: Understanding the concept of limits is important for understanding the behavior of the Fibonacci sequence as it approaches infinity.
The Golden Ratio: The Golden Ratio is an important concept in the Fibonacci sequence because it is involved in the ratio of consecutive Fibonacci numbers. Understanding the properties of the Golden Ratio helps to understand the sequence in a better way.
Binet’s Formula: Binet's formula is an explicit formula that allows us to find out the nth term of the Fibonacci sequence. This formula is derived using the Golden Ratio.
Lucas Sequence: Lucas sequence is like Fibonacci sequence but with different values of the initial terms. This topic helps to understand the relation between Lucas and Fibonacci sequences.
Pascal’s Triangle: Pascal's triangle involves the combination of numbers and plays an important role in generating the Fibonacci sequence.
Fibonnaci Polynomials: Fibonnaci polynomials are a generalization of the Fibonacci sequence. It helps to explore the various properties and patterns of the sequence.
Fibonacci Spiral: Understanding the Fibonacci spiral helps to understand the real-world applications of the Fibonacci sequence, like in Nature, Art, and Architecture.
Fibonacci Heap: The Fibonacci heap is a data structure built upon the Fibonacci sequence. It is used to help in graph algorithms.
Fibonacci Numbers in Nature: The Fibonacci sequence is found in various natural occurrences, such as the growth patterns of plants and the arrangement of leaves in a stem. This topic helps to understand the real-world applications of the Fibonacci sequence.
Fibonacci Sequences in Music and Arts: The Fibonacci sequence is also found in music and art, particularly in the structure and composition of pieces. Understanding this topic helps to appreciate the beauty of the Fibonacci sequence beyond mathematical concepts.
Classic Fibonacci sequence: Starting with 0 and 1, each number in the sequence is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946,...
Lucas numbers: Similar to the Fibonacci sequence, except that the starting values are 2 and 1 rather than 0 and 1: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127,...
Fibonacci prime numbers: Prime numbers in the Fibonacci sequence: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497,...
Fibonacci words: A sequence of binary digits (0's and 1's) with the property that no substring of consecutive digits is repeated: 0, 1, 10, 101, 10110, 10110101, 1011010110110,...
Pisano period: The period of the Fibonacci sequence modulo m, where m is any positive integer. The Pisano period for any value of m is always less than m*m: for m = 10, the period is 60; for m = 100, the period is 300; for m = 1000, the period is 1500, and so on.
Fibonacci polynomials: A polynomial sequence that generalizes the Fibonacci sequence: F(0,x) = 0, F(1,x) = 1, F(n,x) = xF(n-1,x) + F(n-2,x) for n > 1. The first few terms are: F(2,x) = x, F(3,x) = x^2 + 1, F(4,x) = x^3 + 2x, F(5,x) = x^4 + 3x^2 + 1, and so on.
Fibonacci cubes: A sequence of cubes in which each cube is the sum of the previous two: 0^3, 1^3, 1^3 + 0^3, 1^3 + 1^3, 1^3 + 1^3 + 1^3, 1^3 + 1^3 + 2^3, 2^3 + 3^3, 5^3 + 3^3, 8^3 + 5^3, 13^3 + 8^3, and so on.
Fibonacci numbers in nature: The Fibonacci sequence appears in many natural phenomena such as the arrangement of leaves on a stem, the spiral growth of shells and the branching of trees.
Fibonacci spiral: A spiral constructed using the Fibonacci sequence, in which the radius of each consecutive turn is the sum of the radii of the previous two turns.
Fibonacci heap: A data structure used in computer science for priority queue operations, developed by Michael L. Fredman and Robert E. Tarjan in 1984. It is named after Fibonacci because its running time is related to the Fibonacci sequence.
"Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn."
"Starting from 0 and 1, the first few values in the sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144."
"They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci."
"The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry..."
"Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure..."
"They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke..."
"...though they do not occur in all species."
"Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio..."
"...implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases."
"Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences."
"...so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly."
"...on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths."
"...in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths."
"Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure."
"Graphs called Fibonacci cubes used for interconnecting parallel and distributed systems."
"They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and the arrangement of a pine cone's bracts."
"...on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths."
"...also known as Fibonacci..."
"Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences."