Calculators
Centered Pentagonal Number Calculator
Calculate exact centered pentagonal numbers from a ring count, with the next term and digit count, privately in your browser.
Count dots in a centered pentagonal arrangement with n complete rings around one central dot. Enter 0 or a positive whole number of up to 100 digits. Calculations stay in your browser.
Count dots in concentric pentagonal rings.
Start with one central dot and add n complete pentagonal rings. Ring k adds 5k dots, so C(n) = 1 + 5(1 + 2 + … + n) = 5n(n + 1)/2 + 1. For n = 0, 1, 2, 3, 4 the counts are 1, 6, 16, 31, 51. Three rings contain 1 + 5 + 10 + 15 = 31 dots. This calculator uses zero-based ring counts: n = 0 is the central dot alone; references may instead label this term 1. Ordinary pentagonal numbers use n(3n − 1)/2 and count a different arrangement. Enter a non-negative integer of up to 100 digits. Leading zeros and surrounding whitespace are accepted; signs, decimals, separators and scientific notation are rejected. Browser BigInt arithmetic preserves every digit. Results include the next term and decimal digit count. No input is uploaded or stored.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is the formula?
With n rings around one central dot, C(n) = 5n(n + 1)/2 + 1. Ring k adds 5k dots.
What does zero mean?
Zero rings means the central dot alone, so C(0) = 1. The next counts are 6, 16, 31 and 51. Some references label the center as term 1.
Are these ordinary pentagonal numbers?
No. Ordinary pentagonal numbers use n(3n − 1)/2. Centered pentagonal numbers use 5n(n + 1)/2 + 1 to count complete rings around a central dot.
How large can the input be?
Enter a non-negative integer of up to 100 digits. BigInt arithmetic preserves every result digit. Calculations stay in your browser.