Square triangular number: Difference between revisions

Revision as of 20:24, 21 October 2022

Revision as of 20:55, 21 October 2022

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A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers:<ref name=Sylwester>

{{cite journal |lastlast1=Pietenpol |firstfirst1=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 |ISSNissn=0002-9890 |jstor=2312558|publisher=Mathematical Association of America | doi = 10.2307/2312558}}

</ref> If the {{mvar|n}}th triangular number {{math|{{sfrac|''n''(''n'' + 1)|2}}}} is square, then so is the larger {{math|4''n''(''n'' + 1)}}th triangular number, since:

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{{nowrap|1=36 − 1 = 35}}, {{nowrap|1=1225 − 36 = 1189}}, and {{nowrap|1=41616 − 1225 = 40391}}. In other words, the difference between two consecutive square triangular numbers is the square root of another square triangular number.{{citation needed|date=December 2014}}

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 |format=PDF |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>