Centered Polygonal Number¶
A figurate number formed from one central point and successive k-sided dot rings, with one-based count C_k(n)=1+k n(n−1)/2.
Core Idea¶
A centered polygonal number counts dots that start with one center and grow by complete polygonal rings. If the shape has a fixed k sides, each new ring adds k more dots than the preceding ring. With the center called term one, the nth count is 1+k n(n−1)/2. For centered squares (k=4) the sequence begins 1, 5, 13, 25; for centered hexagons (k=6) it begins 1, 7, 19, 37.
Scope of Application¶
This is a family of exact figurate-number sequences in mathematics, not just any drawing with a center. Triangular, square, hexagonal and higher-sided members share the rule. Ordinary polygonal numbers grow from a corner and have different counts. A special identity for one member, such as cube sums of centered hexagonal numbers, is not part of every member's definition.
Clarity¶
The term separates a visual arrangement from the arithmetic sequence and from later theorems about that sequence. It also requires an indexing convention: some sources call the center term one and others call it ring zero, changing the written formula but not the shapes.
Manages Complexity¶
One parameter k, one starting dot, and a ring increment of k(n−1) replace a separate memorized formula for every polygonal family. The same derivation produces all members: add the first n−1 multiples of k to one.
Abstract Reasoning¶
To check a candidate sequence, verify that it starts at one and that successive differences are k,2k,3k,... for one fixed k≥3. Summing these differences gives the closed formula. If a proposed value disagrees, check whether the source uses zero-based indexing before declaring a mathematical contradiction.
Knowledge Transfer¶
The ring-count proof transfers directly between centered triangular, square and hexagonal patterns by changing k. The broad idea of accumulation occurs elsewhere, but the named centered-polygonal identity depends on exact integer dot rings. See the staged V2 for examples, boundary cases and sources.
Neighborhood in Abstraction Space¶
Centered Polygonal Number sits in a sparse region of the domain-specific corpus (82nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Algebraic Substructures & Closures (12 abstractions)
Nearest neighbors
- Circular arc — 0.84
- Ring Ideal — 0.83
- Nine-Point Conic — 0.82
- Polygon — 0.81
- Missing Middle Housing — 0.81
Computed from structural-signature embeddings · 2026-10-08