Pentagonal Number Theorem Calculator

Euler's partition recurrence

All Tools

n (1–100)

p(20) via pentagonal recurrence

627

Generalized Pentagonal Numbers

−ω=1−ω=2+ω=5+ω=7−ω=12−ω=15

Sequence

p(0)1

p(1)1

p(2)2

p(3)3

p(4)5

p(5)7

p(6)11

p(7)15

p(8)22

p(9)30

p(10)42

p(11)56

p(12)77

p(13)101

p(14)135

p(15)176

p(16)231

p(17)297

p(18)385

p(19)490

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About Pentagonal Number Theorem Calculator

A pentagonal number theorem calculator showing Euler's identity Π_{k=1}^∞(1-x^k) = 1 + Σ_{k=1}^∞(-1)^k[x^{k(3k-1)/2}+x^{k(3k+1)/2}]. Used to compute partition numbers via recurrence. Pentagonal numbers: 1,2,5,7,12,15,22... Client-side.

Pentagonal Number Theorem Calculator Features

Pentagonal numbers
Recurrence
p(n) via theorem
Sign pattern
Convergence

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