Toothpick sequence

Count the total line segments in a deterministic planar growth process that starts from one segment and at each stage adds a perpendicular segment at every exposed endpoint.

Version

v2 · 2026-08-30 · History

Domain-specific #

2978

Origin domain

integer sequences

Subdomain

geometric cellular growth

Core Idea

The toothpick sequence records the cumulative number of segments after beginning with one toothpick and, at each synchronous stage, attaching one equal perpendicular toothpick centered at every currently exposed endpoint; the companion first-difference sequence counts additions by stage. Endpoints with degree one form the active frontier, every frontier point generates a perpendicular segment in the next simultaneous update, contacts can close or merge frontier sites, and the cumulative count is obtained by adding each stage's newly created segments.

Its autonomous residual is the exact geometric frontier rule and its induced count sequence, not any toothpick puzzle, arbitrary cellular automaton, visual fractal, or the sequence of newly added pieces unless that convention is declared.

Scope of Application

Toothpick sequence applies when the analyst can specify a staged planar configuration of equal-length horizontal and vertical line segments on a square lattice, together with the set of endpoints incident to exactly one segment and establish that growth begins from the declared seed, updates all and only exposed endpoints synchronously by the perpendicular-centering rule, and records the stated cumulative or stage-increment statistic. The entry locks the standard planar equal-length construction and published indexing relations; variants require distinct rule statements and cannot silently contribute terms.

Clarity

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because the name denotes both the evolving figure and several associated sequences, especially cumulative counts and newly added counts. The disciplined statement is that the object counts as Toothpick sequence exactly when growth begins from the declared seed, updates all and only exposed endpoints synchronously by the perpendicular-centering rule, and records the stated cumulative or stage-increment statistic

Manages Complexity

The abstraction compresses cumulative and first-difference sequences, shifted indexing, finite-stage diagrams, cellular encodings, three-dimensional analogues, and other endpoint-growth rules into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares seed convention, stage index, orientation, endpoint degree, update synchronicity, intersection rule, cumulative total, generation increment, lattice scale, recurrence, and asymptotic growth and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning

1. Type the carrier. Establish a staged planar configuration of equal-length horizontal and vertical line segments on a square lattice, together with the set of endpoints incident to exactly one segment and reject examples from a different problem. 2. Lock the rule. Express that growth begins from the declared seed, updates all and only exposed endpoints synchronously by the perpendicular-centering rule, and records the stated cumulative or stage-increment statistic independently of one notation or implementation.

Knowledge Transfer

Transfer within integer sequences is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from Starting from one horizontal segment, the next stage adds one vertical segment at each of its two exposed ends, producing three total segments; subsequent stages apply the same rule to the newly determined frontier. to The construction can be encoded as a local cellular-growth process whose active sites and segment orientations determine the next finite configuration and hence the increment sequence. demonstrates that continuity.

Relationships to Other Abstractions

Current abstraction Toothpick sequence Domain-specific

Parents (1) — more general patterns this builds on

• Toothpick sequence is a kind of Recurrence Prime

  The proposed strict upward parent is prime:recurrence.

Hierarchy path (1) — routes to 1 parentless root

• Toothpick sequence → Recurrence

Neighborhood in Abstraction Space

Toothpick sequence sits in a sparse region of the domain-specific corpus (69^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Algorithmic Procedures & Discrete Processes (14 abstractions)

Nearest neighbors

• Maze generation algorithm — 0.85
• Pattern formation — 0.85
• Parallel coordinates — 0.84
• Alluvial diagram — 0.84
• Quadrisecant — 0.84

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