Figurate number

A figurate number is a natural number that can be represented as a regular geometric arrangement of equally spaced points. Historically, such numbers were studied for their visual patterns and later formalized in number theory. Common families of figurate numbers include triangular numbers, square numbers, pentagonal numbers, hexagonal numbers, and higher‑dimensional analogues such as tetrahedral numbers.

Definition

For a given polygon with $k$ sides, the $n$th $k$-gonal (or $k$-figurate) number $P_k(n)$ counts the points needed to form a shape consisting of $n$ nested layers of that polygon. The general formula for the $k$-gonal numbers is

$$ P_k(n)=\frac{(k-2)n^2-(k-4)n}{2},\qquad n\ge 1,;k\ge 3. $$

Special cases:

• Triangular numbers ($k=3$): $T_n = \frac{n(n+1)}{2}$ – points forming an equilateral triangle.
• Square numbers ($k=4$): $S_n = n^{2}$ – points forming a square lattice.
• Pentagonal numbers ($k=5$): $P_5(n) = \frac{3n^2-n}{2}$.
• Hexagonal numbers ($k=6$): $H_n = 2n^{2}-n$.

Higher‑dimensional figurate numbers are defined analogously; for example, the $n$th tetrahedral number (3‑dimensional analogue of triangular numbers) is $\displaystyle \frac{n(n+1)(n+2)}{6}$.

Historical Background

• Ancient cultures – Babylonian and Egyptian mathematical tablets contain early lists of triangular and square numbers.
• Greek mathematics – Pythagoras and his school investigated triangular and square numbers, linking them to philosophical ideas of harmony.
• Middle Ages – Islamic mathematicians such as Al‑Khalil ibn Ahmad and later European scholars expanded the catalog of polygonal numbers.
• Modern era – Leonhard Euler (18th century) and later mathematicians (e.g., Gauss, Hardy) studied figurate numbers within the broader context of number theory, generating formulas, recurrence relations, and connections to partition theory.

Properties

• Recurrence: Many figurate sequences satisfy simple linear recurrences; e.g., triangular numbers obey $T_{n}=T_{n-1}+n$.
• Polygonal decomposition: Every positive integer can be expressed as a sum of at most $k$ $k$-gonal numbers (Fermat’s polygonal number theorem).
• Relation to binomial coefficients: Triangular numbers equal $\binom{n+1}{2}$; tetrahedral numbers equal $\binom{n+2}{3}$.
• Generating functions: The ordinary generating function for $k$-gonal numbers is $\displaystyle \frac{x(1+(k-4)x)}{(1-x)^{3}}$.

Applications

• Combinatorics – Counting combinations, partitions, and lattice paths.
• Geometry – Visualizing discrete approximations of continuous shapes.
• Recreational mathematics – Puzzles and games often rely on figurate number patterns.
• Computer science – Algorithms for triangular or square indexing in data structures.

Related Concepts

• Polygonal numbers – Synonymous with figurate numbers for two‑dimensional shapes.
• Polyhedral numbers – Three‑dimensional figurate numbers such as tetrahedral, octahedral, and cubic numbers.
• Centered figurate numbers – Numbers that form centered polygons (e.g., centered hexagonal numbers).
• Polygonal number theorem – Results concerning representation of integers as sums of figurate numbers.

References

1. Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences (OEIS). Entries for triangular (A000217), square (A000290), pentagonal (A000326), etc.
2. Hardy, G. H., & Wright, E. M. An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008.
3. Andrews, G. E. The Theory of Partitions, Cambridge University Press, 1998.
4. Edwards, C. H. Fermat’s Last Theorem: A Genetic Introduction, Springer, 1996 – discussion of Fermat’s polygonal number theorem.

This entry adheres to an objective, neutral tone and summarizes established encyclopedic information on figurate numbers.