Fibonacci sequence

The sequence 0, 1, 1, 2, 3, 5, 8, … in which each term after the first two is the sum of the preceding two.

Recurrence relation: An equation that defines each term of a sequence using earlier terms. The rule adding the two previous terms is a recurrence relation.
Integer sequence: An ordered list of integers indexed by whole-number positions. The Fibonacci sequence is one specific ordered list of integers.
Pingala: An ancient Indian scholar associated with an early system for analyzing poetic meters. Patterns in short and long syllables yield counts that follow the Fibonacci recurrence.
Fibonacci coding: A variable-length binary code that represents positive integers using sums of nonconsecutive Fibonacci numbers. The sequence's additive structure supplies the code's representation system.
Lucas numbers: The sequence 2, 1, 3, 4, 7, … defined by the Fibonacci recurrence with initial values 2 and 1. They share the recurrence but differ in starting values and resulting terms.
Generating function: A formal power series whose coefficients encode a sequence. The Fibonacci numbers are encoded by the rational function x/(1−x−x²).
Mathematical induction: A proof method that establishes a base case and then proves each case follows from its predecessor. Induction proves identities about all Fibonacci numbers from the recurrence.
Virahanka: An Indian prosodist who described counting patterns for combinations of short and long syllables. His account predates Fibonacci's and contains the sequence's recurrence pattern.
Zeckendorf's theorem: The theorem that every positive integer has a unique representation as a sum of nonconsecutive Fibonacci numbers. It gives a canonical way to express integers using the sequence.
Tribonacci sequence: A sequence in which each term after the first three is the sum of the preceding three. It generalizes the same additive rule using three preceding terms.

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