Golomb sequence¶
The nondecreasing self-describing integer sequence in which each positive integer n occurs exactly as many times as the nth term states.
Core Idea¶
With the initial value one and least-positive continuation convention, the self-frequency constraint determines a unique sequence whose run lengths recursively reproduce its own terms. Each completed run reports how many copies of its index must appear, and the minimal monotone continuation satisfies the next required frequency without violating earlier counts. 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¶
Golomb sequence belongs to integer sequences and is useful where the analyst can specify the typed integer sequences carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the indexing origin, positive-integer range, monotonicity, initial value, self-frequency equation, least-continuation convention and any recurrence or asymptotic statement are explicit. The scope is broad within that domain but bounded by the need for the indexing origin, positive-integer range, monotonicity, initial value, self-frequency equation, least-continuation convention and any recurrence or asymptotic statement are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the indexing origin, positive-integer range, monotonicity, initial value, self-frequency equation, least-continuation convention and any recurrence or asymptotic statement 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. A bare label is insufficient because the name Golomb sequence can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
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 Golomb sequence. Golomb 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 integer sequences carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the indexing origin, positive-integer range, monotonicity, initial value, self-frequency equation, least-continuation convention and any recurrence or asymptotic statement are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of integer sequences because they reuse the typed integer sequences carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Each completed run reports how many copies of its index must appear, and the minimal monotone continuation satisfies the next required frequency without violating earlier counts., and type the carrier, state every parameter and convention in the definition, test that the indexing origin, positive-integer range, monotonicity, initial value, self-frequency equation, least-continuation convention and any recurrence or asymptotic statement are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Golomb sequence Domain-specific
Parents (1) — more general patterns this builds on
-
Golomb sequence is a kind of Reflexivity (Self-Reference) Prime
The proposed strict upward parent is
prime:reflexivity_self_reference.
Hierarchy path (1) — routes to 1 parentless root
- Golomb sequence → Reflexivity (Self-Reference)
Neighborhood in Abstraction Space¶
Golomb sequence sits in a moderately populated region (57th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Complete sequence — 0.88
- Generalized taxicab number — 0.88
- Lambek–Moser theorem — 0.87
- Recurrence relation — 0.87
- Schröder number — 0.87
Computed from structural-signature embeddings · 2026-09-08