Constant-recursive sequence¶

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

Version: v1 · 2026-09-08 · History
Domain-specific #: 3858
Origin domain: discrete mathematics
Subdomain: discrete 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.

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. 2. 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) — more general patterns this builds on

Constant-recursive sequence is a kind of Recursion Prime

The proposed strict upward parent is prime:recursion.

Hierarchy path (1) — routes to 1 parentless root

Constant-recursive sequence → Recursion

Neighborhood in Abstraction Space¶

Constant-recursive sequence sits in a crowded region of the domain-specific corpus (11^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Recurrences & Integer Sequences (5 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08