Editing Square triangular number (section)

==Other characterizations==

All square triangular numbers have the form <math>b^2c^2</math>, where <math>\tfrac{b}{c}</math> is a [[Convergent (continued fraction)|convergent]] to the [[simple continued fraction|continued fraction expansion]] of <math>\sqrt2</math>, the [[square root of 2]].<ref name=Ball>
{{cite book | last1 = Ball | first1 = W. W. Rouse |author-link1 = W. W. Rouse Ball | last2 = Coxeter | first2 = H. S. M. |author-link2 = Harold Scott MacDonald Coxeter | title = Mathematical Recreations and Essays | url = https://archive.org/details/mathematicalrecr00coxe | url-access = limited | publisher = Dover Publications | location = New York | year = 1987 | page = [https://archive.org/details/mathematicalrecr00coxe/page/n72 59]| isbn = 978-0-486-25357-2 }}
</ref>

A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the <math>n</math>th triangular number <math>\tfrac{n(n+1)}{2}</math> is square, then so is the larger <math>4n(n+1)</math>th triangular number, since:

{{bi|left=1.6|<math>\displaystyle\frac{\bigl( 4n(n+1) \bigr) \bigl( 4n(n+1)+1 \bigr)}{2} = 4 \, \frac{n(n+1)}{2} \,\left(2n+1\right)^2.</math>}}

The left hand side of this equation is in the form of a triangular number, and as the product of three squares, the right hand side is square.<ref name=Sylwester>
{{cite journal |last1=Pietenpol |first1=J. L. |first2=A. V. |last2=Sylwester |first3=Erwin |last3=Just |first4=R. M. |last4=Warten  |date=February 1962 |title=Elementary Problems and Solutions: E 1473, Square Triangular Numbers |journal=American Mathematical Monthly |volume=69 |issue=2 |pages=168–169 |issn=0002-9890  |jstor=2312558|publisher=Mathematical Association of America | doi = 10.2307/2312558}}
</ref>

The [[generating function]] for the square triangular numbers is:<ref>{{cite web |first=Simon |last=Plouffe |author-link=Simon Plouffe |title=1031 Generating Functions |url=http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf |publisher=University of Quebec, Laboratoire de combinatoire et d'informatique mathématique |page=A.129 |date=August 1992 |access-date=2009-05-11 |archive-date=2012-08-20 |archive-url=https://web.archive.org/web/20120820012535/http://www.plouffe.fr/simon/articles/FonctionsGeneratrices.pdf |url-status=dead }}</ref>
:<math>\frac{1+z}{(1-z)\left(z^2 - 34z + 1\right)} = 1 + 36z + 1225 z^2 + \cdots</math>