07-07-2026, 08:00 PM
Constant-recursive sequence
Summary
A constant-recursive sequence, also known as a linear recurrence sequence, is an infinite sequence in which every term is determined by a fixed linear combination of a finite number of preceding terms. One of the best-known examples is the Fibonacci sequence, where each number equals the sum of the previous two, but the concept also includes arithmetic progressions, geometric progressions, polynomial sequences, and many other familiar mathematical patterns.
These sequences are defined by linear recurrence relations with constant coefficients, allowing complex numerical behavior to be described through relatively simple formulas. However, not every sequence fits this framework—well-known examples such as the factorial and Catalan sequences fall outside the class because their growth and structure cannot be captured by a constant-coefficient recurrence.
Beyond their elegant definitions, constant-recursive sequences play an important role across mathematics and computer science. They appear in combinatorics, algebraic number theory, finite differences, algorithm analysis, and formal language theory, where they help model recursive processes, count combinatorial objects, and describe computational complexity. They also possess powerful mathematical properties, remaining stable under operations such as addition, multiplication, and generating-function transformations.
Deep results, including the Skolem–Mahler–Lech theorem, reveal surprising regularity in the positions of zero terms, while the still-unsolved Skolem problem highlights that fundamental questions about these sequences remain open. Together, these properties make constant-recursive sequences a cornerstone of discrete mathematics, linking elegant theoretical ideas with practical applications in algorithms, computation, and mathematical modeling.
ARTICLE
Summary
A constant-recursive sequence, also known as a linear recurrence sequence, is an infinite sequence in which every term is determined by a fixed linear combination of a finite number of preceding terms. One of the best-known examples is the Fibonacci sequence, where each number equals the sum of the previous two, but the concept also includes arithmetic progressions, geometric progressions, polynomial sequences, and many other familiar mathematical patterns.
These sequences are defined by linear recurrence relations with constant coefficients, allowing complex numerical behavior to be described through relatively simple formulas. However, not every sequence fits this framework—well-known examples such as the factorial and Catalan sequences fall outside the class because their growth and structure cannot be captured by a constant-coefficient recurrence.
Beyond their elegant definitions, constant-recursive sequences play an important role across mathematics and computer science. They appear in combinatorics, algebraic number theory, finite differences, algorithm analysis, and formal language theory, where they help model recursive processes, count combinatorial objects, and describe computational complexity. They also possess powerful mathematical properties, remaining stable under operations such as addition, multiplication, and generating-function transformations.
Deep results, including the Skolem–Mahler–Lech theorem, reveal surprising regularity in the positions of zero terms, while the still-unsolved Skolem problem highlights that fundamental questions about these sequences remain open. Together, these properties make constant-recursive sequences a cornerstone of discrete mathematics, linking elegant theoretical ideas with practical applications in algorithms, computation, and mathematical modeling.
ARTICLE
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│ KONSTANTINOS MICHAILIDIS │
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