Integer sequence

An ordered list of integers indexed by whole-number positions. Its terms may follow a rule, arise from a process, or be studied without a known formula.

Sequence: An ordered collection of elements indexed by positions, often defined by a rule or recurrence. An integer sequence is the integer-valued case of this broader structure.
Arithmetic progression: A sequence whose consecutive terms differ by a fixed constant. Its integer terms arise by repeatedly adding a fixed integer.
Fibonacci sequence: The sequence beginning 0, 1 in which each later term is the sum of the previous two. It is a canonical example defined by a simple recurrence.
Combinatorics: The branch of mathematics concerned with counting, arrangement, and discrete structures. Many integer sequences are counts of combinatorial objects.
Sequence extrapolation: The prediction of later terms from a finite initial segment of a sequence. The same initial integer terms can fit many incompatible continuation rules.
Integer: A whole number, positive, negative, or zero. Every term in an integer sequence belongs to this number system.
Geometric progression: A sequence whose consecutive terms have a fixed ratio. Integer-valued examples result when repeated multiplication preserves integrality.
Prime-counting function: The function that counts primes less than or equal to a given number. Its values form an integer sequence recording the distribution of primes.
Enumerative combinatorics: The study of counting discrete structures according to specified properties. It produces sequences whose terms count objects of each size.
Collatz conjecture: The conjecture that iterating a simple parity-based rule from any positive integer eventually reaches 1. Its iteration creates integer sequences whose long-term behavior remains unproved.

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