Pentagonal Number Calculator
Pentagonal numbers are figurate numbers that count dots arranged in nested pentagons sharing a common corner, with the sequence beginning 1, 5, 12, 22, 35. The nth pentagonal number is given by the exact formula P(n) = n(3n - 1) / 2, which this calculator evaluates in a single step. Pentagonal numbers are central to Euler's pentagonal number theorem in the study of integer partitions. Enter a positive whole number index n to get the pentagonal number, the next term in the sequence, and the gap between them.
Pentagonal number formula
P(n) = n * (3n - 1) / 2
P(n+1) = (n + 1) * (3n + 2) / 2
Step = P(n+1) - P(n) = 3n + 1
The closed form gives the pentagonal number directly. The gap between consecutive pentagonal numbers grows linearly as 3n + 1, which is why the sequence accelerates faster than triangular numbers.
About pentagonal numbers
- The sequence begins 1, 5, 12, 22, 35, 51, 70, 92, 117, 145.
- The difference between successive pentagonal numbers is 4, 7, 10, 13, increasing by three each time.
- Euler's pentagonal number theorem uses generalised pentagonal numbers to compute the partition function.
- Pentagonal numbers are figurate numbers, the same family as triangular, square and hexagonal numbers.
- For n = 1 the pentagonal number is 1, the seed shared by every figurate sequence.
Pentagonal numbers: frequently asked questions
What is a pentagonal number?
A pentagonal number counts the dots in a pattern of nested pentagons sharing a corner. The nth pentagonal number is given by n(3n - 1) / 2, and the sequence begins 1, 5, 12, 22, 35.
What is the formula for the nth pentagonal number?
P(n) = n times (3n - 1) divided by 2. For n = 3 this gives 3 times 8 divided by 2, which equals 12.
How do pentagonal numbers differ from triangular numbers?
Triangular numbers use the formula n(n + 1) / 2 and grow more slowly. Pentagonal numbers use n(3n - 1) / 2 and grow faster because of the factor of three on the leading term.
What are generalised pentagonal numbers?
Generalised pentagonal numbers allow n to be zero or negative as well as positive, producing 0, 1, 2, 5, 7, 12, 15. They appear in Euler's pentagonal number theorem for partitions. This calculator uses positive integer n for the standard sequence.
Where do pentagonal numbers appear?
They appear in number theory, especially Euler's pentagonal number theorem linking them to partition functions, and as figurate numbers describing dot patterns shaped like pentagons.
Official sources
- NIST Digital Library of Mathematical Functions: Integer Partitions: Other.
- NIST Digital Library of Mathematical Functions: Functions of Number Theory.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.