Pentagonal numbers: 1, 5, 12, 22, 35, 51, 70... P(n) = n(3n−1)/2. Generalized: P(−n) = n(3n+1)/2 gives 0, 2, 7, 15, 26... Euler's pentagonal theorem: Π(1−x^n) = Σ (−1)^k x^{P(k)} gives a recurrence for the partition function p(n).
How to Use
Enter n or test a value:
- P(n): nth pentagonal
- Test: Is N pentagonal?
- Generalized: Both signs
Euler's Theorem
Π_{n≥1}(1−x^n) = 1 − x − x² + x⁵ + x⁷ − x¹² − x¹⁵ + ... The exponents are generalized pentagonal numbers! This gives the fastest recurrence for computing p(n): p(n) = Σ (−1)^{k+1} p(n−P(k)).
Geometric Meaning
Arrange dots in nested pentagons: 1, 5, 12, 22... Each layer adds a larger pentagon ring. The formula n(3n−1)/2 counts dots. Unlike triangular/square numbers, pentagonal numbers have deep connections to partition theory.
Step-by-Step Instructions
- 1Enter n.
- 2Compute P(n).
- 3Generate sequence.
- 4Test membership.
- 5See Euler connection.