gives the n triangular number .
gives the n r-gonal number .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- PolygonalNumber[n] is generically defined as .
- PolygonalNumber[r,n] is generically defined as .
- PolygonalNumber[r,n] can be interpreted as the number of points arranged in the form of n-1 polygons of r sides. For instance for r=3 and n=4:
- PolygonalNumber automatically threads over lists.
Examplesopen allclose all
Basic Examples (2)
Use PolygonalNumber with integer arguments:
Use RegularPolygon to specify the number of sides of a regular polygon:
PolygonalNumber automatically threads over lists:
Use PolygonalNumber with symbolic input:
Properties & Relations (9)
The 0 polygonal number of any regular polygon is zero:
The first polygonal number of any regular polygon is one:
Every other triangular number is a hexagonal number:
The sum of two consecutive triangular numbers is a square number:
Every pentagonal number is one third of a triangular number:
The difference between the n r-gonal number and the n (r+1)-gonal number is the (n-1) triangular number:
Even perfect numbers are triangular numbers related to Mersenne prime exponents:
Even perfect numbers are hexagonal numbers related to Mersenne prime exponents:
All even perfect numbers greater than 6 are of the following form for some value of k:
Wolfram Research (2016), PolygonalNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/PolygonalNumber.html.
Wolfram Language. 2016. "PolygonalNumber." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PolygonalNumber.html.
Wolfram Language. (2016). PolygonalNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolygonalNumber.html