Nº Pentagonal n
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
P_n = n(3n−1)/2
About this calculator
The Pentagonal Number n Calculator computes the nth pentagonal number, defined by the formula P_n = n(3n−1)/2. Pentagonal numbers are figurate numbers that represent pentagons, with each layer adding points. For example, P_1 = 1, P_2 = 5, P_3 = 12, P_4 = 22, and so on. The calculator accepts a positive integer n and returns the corresponding pentagonal number.
This tool is useful for math students exploring numerical sequences, geometric patterns, and properties of figurate numbers. It can also be used in combinatorics or number theory problems, such as representing numbers as sums of two pentagonal numbers. The calculation is instant and accurate; simply enter the value of n.
Caveats: the formula is valid only for positive integers n (n ≥ 1). For n = 0, the result would be 0, but the traditional sequence starts at n = 1. Ensure the input is an integer. The calculator does not accept negative numbers or decimals, as the definition of pentagonal numbers applies to natural indices.
Frequently asked questions
What is a pentagonal number?
A pentagonal number is a figurate number that represents a pentagon of dots. The nth pentagonal number is given by the formula P_n = n(3n−1)/2.
How do I use this calculator?
Enter a positive integer n in the input field and click Calculate. The result will be the nth pentagonal number.
Can I use negative or decimal numbers?
No, the calculator only accepts positive integers. Negative numbers or decimals are not valid for the pentagonal sequence.
What is the first pentagonal number?
The first pentagonal number is P_1 = 1. The sequence starts at n = 1.
What are pentagonal numbers used for?
They are used in mathematics to study geometric patterns, in number theory, and in combinatorics problems, such as representing numbers as sums of two pentagonal numbers.