A figurate number, also (but mostly in texts from the 1500 and 1600s) known as a figural number (Simpson and Weiner 1992, p. 587), is a number that can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a regular polygon, the number is called a polygonal number. The polygonal numbers illustrated above are called triangular, square, pentagonal, and hexagonal numbers, respectively. Figurate numbers can also form other shapes such as centered polygons, L-shapes, three-dimensional solids, etc.
The 
th
regular 
-polytopic
number is given by
where 
is the multichoose function, 
is a binomial coefficient,
and 
is a rising factorial. Special cases therefore
include the triangular numbers
 |
(4) |
 |
(5) |
 |
(6) |
and so on (Dickson 2005, p. 7).
The following table lists the most common types of figurate numbers.
See also
Biquadratic Number, Centered Cube Number, Centered Pentagonal Number, Centered Polygonal Number, Centered Square Number, Centered Triangular Number, Cubic Number, Decagonal Number, Figurate Number Triangle, Gnomonic Number, Heptagonal Number, Heptagonal Pyramidal Number, Hex Number, Hex Pyramidal Number, Hexagonal Number, Hexagonal Pyramidal Number, Multichoose, Nexus Number, Octagonal Number, Octahedral Number, Pentagonal Number, Pentagonal Pyramidal Number, Pentatope Number, Polygonal Number, Pronic Number, Pyramidal Number, Rhombic Dodecahedral Number, Square Number, Square Pyramidal Number, Stella Octangula Number, Tetrahedral Number, Triangular Number, Truncated Octahedral Number, Truncated Tetrahedral Number
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References
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 30-62, 1996.Dickson, L. E. "Polygonal, Pyramidal, and Figurate Numbers." Ch. 1 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 1-39, 2005.Goodwin, P. "A Polyhedral Sequence of Two." Math. Gaz. 69, 191-197, 1985.Guy, R. K. "Figurate Numbers." §D3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 147-150, 1994.Kraitchik, M. "Figurate Numbers." §3.4 in Mathematical Recreations. New York: W. W. Norton, pp. 66-69, 1942.Savin, A. "Shape Numbers." Quantum 11, 14-18, 2000.Simpson, J. A. and Weiner, E. S. C. (Preparers). The Compact Oxford English Dictionary, 2nd ed. Oxford, England: Clarendon Press, 1992.
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Cite this as:
Weisstein, Eric W. "Figurate Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FigurateNumber.html