Square triangular number: Difference between revisions

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

{{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> 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:

:{{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>}}

As the product of three squares, the right hand side is square. The triangular roots {{math|''t_k''}} are alternately simultaneously one less than a square and twice a square if {{mvar|k}} is even, and simultaneously a square and one less than twice a square if {{mvar|k}} is odd. Thus,

:49 = 7² = 2 × 5² − 1,

:288 = 17² − 1 = 2 × 12², and

:1681 = 41² = 2 × 29² − 1.

In each case, the two square roots involved multiply to give {{math|''s_k''}}: {{nowrap|1=5 × 7 = 35}}, {{nowrap|1=12 × 17 = 204}}, and {{nowrap|1=29 × 41 = 1189}}.{{citation needed|date=December 2014}}

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>

Additionally:

{{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}}

:<math>N_k - N_{k-1}=s_{2k-1};</math>

</ref>

{{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 |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>