Calculators
Pentagonal Number Calculator
Calculate the nth pentagonal number exactly with a formula breakdown, privately in your browser.
Count dots in a pentagonal arrangement.
The nth ordinary pentagonal number is P(n) = n × (3n − 1) ÷ 2. The first five positive values are 1, 5, 12, 22, and 35. Successive layers add 1, 4, 7, 10, and so on, increasing by 3 each time. For n = 10, P(10) = 10 × 29 ÷ 2 = 145. We include P(0) = 0 for an empty arrangement. Enter a non-negative index of up to 100 digits without separators, signs, decimals or scientific notation. Leading zeros are accepted. Browser BigInt arithmetic keeps the entire result exact. This tool calculates ordinary pentagonal numbers from their indices; it does not calculate centered or generalized pentagonal numbers or test sequence membership. All processing stays in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is a pentagonal number?
An ordinary pentagonal number counts dots arranged in a pentagonal pattern. The sequence starts 1, 5, 12, 22, 35. Successive differences are 4, 7, 10, 13, increasing by 3.
How is the nth pentagonal number calculated?
Use P(n) = n × (3n − 1) ÷ 2. For n = 10, this gives 10 × 29 ÷ 2 = 145. The numerator is always even for integer n.
Are these centered pentagonal numbers?
No. Ordinary pentagonal numbers use n × (3n − 1) ÷ 2. Centered pentagonal numbers follow a different formula, and generalized pentagonal numbers also allow negative indices. This calculator uses ordinary non-negative indices only.
Can I enter zero or decimal indices?
Zero returns zero for an empty arrangement. Negative and fractional indices are rejected. Enter digits only without signs, commas or exponent notation; leading zeros are accepted.
Are large results rounded or uploaded?
No. BigInt arithmetic produces exact integers for indices up to 100 digits. Calculations stay in your browser, and you can copy the full result.