Constant-recursive sequence¶

A sequence satisfying a fixed finite-order linear recurrence with constant coefficients.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3858
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Discrete Mathematics → Mathematics
Aliases: Linear recurrence sequence, C-finite sequence

Core Idea¶

Initial values, coefficient field, order and validity range are constitutive, zero leading coefficients can lower the order and nonhomogeneous recurrences are a different class unless augmented. A finite state of previous terms is multiplied by fixed coefficients to generate the next term, equivalently powers of a companion matrix or a rational generating function determine the sequence. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

The Same-Recipe Number Train

Imagine a line of numbers where you make each new number from the few numbers just before it, using the very same recipe every time. Like: start with 1 and 1, then always add the last two, so you get 1, 1, 2, 3, 5, 8. You need the starting numbers and the recipe, and then the whole line builds itself.

Numbers From the Last Few

A constant-recursive sequence is a list of numbers made by a fixed rule. You remember the last few numbers, multiply each one by a number that never changes, and add them up to get the next number. For example, 'next = last + the one before it' starting from 1 and 1 gives 1, 1, 2, 3, 5, 8, 13. To pin down the sequence you have to say both the starting numbers and the rule, because the same rule with different starts gives a different list.

Fixed Linear Recurrence Sequence

A constant-recursive sequence is defined by a rule like a(n) = c1·a(n−1) + c2·a(n−2) + … + cd·a(n−d), where the coefficients c1…cd are fixed constants and d is the order. To pin the sequence down you must also give the first d values and say where the index starts. The rule is homogeneous: there is no extra added term, so a rule like a(n) = 2a(n−1) + 1 belongs to a different class unless you rewrite it with an extra state. The same sequence can be described by a characteristic polynomial, which gives a closed-form formula for the nth term. Contrast this with a sequence whose rule changes coefficients as n grows; that is not constant-recursive.

A constant-recursive (C-finite) sequence satisfies a homogeneous linear recurrence a(n+k) = c1 a(n+k-1) + ... + ck a(n) with constant coefficients in a stated field or ring, valid from a stated start index. Its identity requires declaring the coefficient field, the indexing, the order k and the k initial values. Equivalently, the state vector of the last k terms is advanced by powers of a fixed companion matrix, and the ordinary generating function is a rational function P(x)/Q(x) whose denominator encodes the recurrence. The characteristic polynomial of the recurrence gives closed forms as sums of polynomial-times-exponential terms in its roots. A sequence may satisfy many recurrences; the one of least order is the minimal recurrence, and a vanishing leading coefficient can drop the effective order. Nonhomogeneous recurrences (with a forcing term) are a separate class unless augmented into a larger homogeneous system.

Scope of Application¶

Constant-recursive sequence belongs to discrete mathematics and is useful where the analyst can specify the typed discrete mathematics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the scalar ring or field, sequence index and initial values, recurrence order, constant coefficients, homogeneous linear recurrence equation and start index, minimal recurrence, companion matrix, characteristic polynomial and closed-form and rational-generating-function equivalences are explicit. The scope is broad within that domain but bounded by the need for the scalar ring or field, sequence index and initial values, recurrence order, constant coefficients, homogeneous linear recurrence equation and start index, minimal recurrence, companion matrix, characteristic polynomial and closed-form and rational-generating-function equivalences are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the scalar ring or field, sequence index and initial values, recurrence order, constant coefficients, homogeneous linear recurrence equation and start index, minimal recurrence, companion matrix, characteristic polynomial and closed-form and rational-generating-function equivalences are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Constant-recursive sequence. Constant-recursive sequence compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed discrete mathematics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the scalar ring or field, sequence index and initial values, recurrence order, constant coefficients, homogeneous linear recurrence equation and start index, minimal recurrence, companion matrix, characteristic polynomial and closed-form and rational-generating-function equivalences are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of discrete mathematics because they reuse the typed discrete mathematics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, A finite state of previous terms is multiplied by fixed coefficients to generate the next term, equivalently powers of a companion matrix or a rational generating function determine the sequence., and type the carrier, state every parameter and convention in the definition, test that the scalar ring or field, sequence index and initial values, recurrence order, constant coefficients, homogeneous linear recurrence equation and start index, minimal recurrence, companion matrix, characteristic polynomial and closed-form and rational-generating-function equivalences are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Constant-recursive sequence Domain-specific

Parents (1) — outgoing typed relations

Constant-recursive sequence is a kind of Recursion Prime

It remains its own entry because its identity is fixed by the scalar ring or field, sequence index and initial values, recurrence order, constant coefficients, homogeneous linear recurrence equation and start index, minimal recurrence, companion matrix, characteristic polynomial and closed-form and rational-generating-function equivalences.

Hierarchy path (1) — routes to 1 parentless root

Constant-recursive sequence → is a kind of Recursion

Neighborhood in Abstraction Space¶

Constant-recursive sequence sits in a crowded region of the domain-specific corpus (7^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Numeral Systems & Integer Sequences (22 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25