Pentagonal Number Theorem

From the previous problems, we have learnt that the Pentagonal Numbers are given by the formula pn=3n2n2 p_n = \frac{ 3n^2- n} { 2} . We have investigated the sequence for positive integers nn, which gives us the values:

1,5,12,22,35,51, 1, 5, 12, 22, 35, 51, \ldots

What happens when we substitute in non-positive integers into this formula? We will get a different sequence, namely:

0,2,7,15,26,40,57, 0, 2, 7, 15, 26, 40, 57, \ldots

Together, these numbers are called the Generalized Pentagonal Numbers. They appear in the following theorem:

The Pentagonal Number Theorem states that
n=1(1xn)=k=(1)kxpk=1xx2+x5+x7x12x15 \prod_{n=1}^\infty ( 1 - x^n) = \sum_{k=-\infty}^{\infty} (-1)^k x^{p_k} = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} \ldots

If we take the combinatorial interpretation of the identity, the LHS is the generating function for the number of partitions of nn into an even number of distinct parts, minus the number of partitions of nn into an odd number of distinct parts. By looking at the coefficients on the RHS, the surprising corollary is that this difference is either -1, 0 or 1 always!

The combinatorial proof of the Pentagonal Number Theorem follows a similar argument. It creates a bijection between the number of even parts, and the number of odd parts, and then accounts for the cases where the bijection fails, namely at the values pk=3k2k2 p_k = \frac{ 3k^2 - k } { 2 } . You can attempt to prove it for yourself, or read the Wikipedia article.

Note by Calvin Lin
5 years, 6 months ago

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nice

Hemant Maheshwari - 5 years, 6 months ago

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Does it work for n-gon as well? I haven't tried to generalize it yet........

Maharnab Mitra - 5 years, 6 months ago

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Does what work for n-gon?

If you're referring to finding the formula for the number of dots in Figure K of an N-gon diagram, then yes you can use similar approaches. The easiest will be to find the Arithmetic Progression, and then find the sum of that. I wanted to show a different approach, which uses the idea of Mathematical Induction, or Method of Differences.

Calvin Lin Staff - 5 years, 6 months ago

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I meant the number of dots only. Thanks, sir.

Maharnab Mitra - 5 years, 6 months ago

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@Maharnab Mitra Why don't you investigate and then write up a note? I'd be interested in your results.

Calvin Lin Staff - 5 years, 6 months ago

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great

Shishir Das - 5 years, 6 months ago

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Greaat :) It was very interesting. Thank you !

Math Nerd - 5 years, 6 months ago

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it would be always positive integer.

Kazi Mamunar Rashid - 5 years, 6 months ago

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i will continue my thinking unless i will got it. Thanks

Md. Zahidul Islam - 5 years, 5 months ago

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Fantastic

Hatim Baroodwala - 5 years, 4 months ago

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Nice one sir! Can I suggest something? Knowing 1st pentagonal number is one, we can see that number of dots to be added to form the next pentagonal number are in A.P. - where a=4, and common difference d=3.... Thus can't we express pentagonal numbers as sums of the increasing A.P. 1,4,7,10; i.e. S1, S2, S3 and so on, where S(n) is the sum to 'n' terms of the A.P. 1,4,7,10...?

Ravi Mistry - 5 years, 1 month ago

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