Square triangular number: Difference between revisions

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[[File:square_triangular_number_36.svg|234px|thumb|Square triangular number 36 depicted as a triangular number and as a square number.]]

In [[mathematics]], a '''square triangular number''' (or '''triangular square number''') is a number which is both a [[triangular number]] and a [[square number]], in other words, the sum of all integers from <math>1</math> to <math>n</math> has a square root that is an integer. There are [[Infinity|infinitely many]] square triangular numbers; the first few are:

{{bi|left=1.6|0, 1, 36, {{val|1225}}, {{val|41616}}, {{val|1413721}}, {{val|48024900}}, {{val|1631432881}}, {{val|55420693056}}, {{val|1882672131025}} {{OEIS|id=A001110}}}}

==Solution as a Pell equation==

==Explicit formulas==

Write <math>N_k</math> for the <math>k</math>th square triangular number, and write <math>s_k</math> and <math>t_k</math> for the sides of the corresponding square and triangle, so that

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{{bi|left=1.6|<math>\displaystyle N_k = s_k^2 = \frac{t_k(t_k+1)}{2}.</math>}}

Define the ''triangular root'' of a triangular number <math>N=\tfrac{n(n+1)}{t2}</math> to be <math>n</math>. From this definition and the quadratic formula,

{{bi|left=1.6|<math>\displaystyle n = \frac{\sqrt{8N + 1} - 1}{2}.</math>}}

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{{bi|left=1.6|<math>\displaystyle s_k = y_k , \quad t_k = \frac{x_k - 1}{2}, \quad N_k = y_k^2.</math>}}

Hence, the first square triangular number, derived from <math>(3,1)</math>, is <math>1</math>, and the next, derived from <math>6\cdot (3,1)-(1,0)-=(17,6)</math>, is <math>36</math>.

The sequences <math>N_k</math>, <math>s_k</math> and <math>t_k</math> are the [[OEIS]] sequences {{OEIS2C|id=A001110}}, {{OEIS2C|id=A001109}}, and {{OEIS2C|id=A001108}} respectively.

==Explicit formula==

In 1778 [[Leonhard Euler]] determined the explicit formula<ref name=Dickson>

{{cite book | last1 = Dickson | first1 = Leonard Eugene | author-link1 = Leonard Eugene Dickson |title = [[History of the Theory of Numbers]] | volume = 2 | publisher = American Mathematical Society | ___location = Providence | year = 1999 |orig-year = 1920 | page = 16 | isbn = 978-0-8218-1935-7 }}

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==Recurrence relations==

The solution to the Pell equation can be expressed as a [[recurrence relation]] for the equation's solutions. This can be translated into recurrence equations that directly express the square triangular numbers, as well as the sides of the square and triangle involved. We have<ref>{{MathWorld|title=Square Triangular Number|urlname=SquareTriangularNumber}}</ref>{{Rp|(12)}}

There are [[recurrence relation]]s for the square triangular numbers, as well as for the sides of the square and triangle involved. We have<ref>{{MathWorld|title=Square Triangular Number|urlname=SquareTriangularNumber}}</ref>{{Rp|(12)}}

{{bi|left=1.6|<math>\displaystyle \begin{align}

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==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>

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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>

==Numerical data==

As {{mvar|k}} becomes larger, the ratio {{math|{{sfrac|''t_k''|''s_k''}}}} approaches [[square root of 2|{{sqrt|2}}]] ≈&nbsp;{{val|1.41421356}}, and the ratio of successive square triangular numbers approaches {{nowrap|(1 + {{sqrt|2}})⁴}} {{nowrap|{{=}} 17 + 12{{sqrt|2}}}} ≈&nbsp;{{val|33.970562748}}. The table below shows values of {{mvar|k}} between 0 and 11, which comprehend all square triangular numbers up to {{val|e=16}}.

:{| class="wikitable" border="1" style="text-align:right"

|-

! {{mvar|k}}

! {{math|''N_k''}}

! {{math|''s_k''}}

! {{math|''t_k''}}

!rowspan=2 valign=top| {{math|{{sfrac|''t_k''|''s_k''}}}}

!rowspan=3 valign=top| {{math|{{sfrac|''N_k''|''N''_{''k'' − 1}}}}}

|-

|0

|0

|0

|0

|-

|1

|1

|1

|1

|align=left|1

|-

|2

|36

|6

|8

|align=left|{{val|1.33333333}}

|align=left|36

|-

|3

|{{val|1225|fmt=gaps}}

|35

|49

|align=left|1.4

|align=left|{{val|34.027777778}}

|-

|4

|{{val|41616}}

|204

|288

|align=left|{{val|1.41176471}}

|align=left|{{val|33.972244898}}

|-

|5

|{{val|1413721}}

|{{val|1189|fmt=gaps}}

|{{val|1681|fmt=gaps}}

|align=left|{{val|1.41379310}}

|align=left|{{val|33.970612265}}

|-

|6

|{{val|48024900}}

|{{val|6930|fmt=gaps}}

|{{val|9800|fmt=gaps}}

|align=left|{{val|1.41414141}}

|align=left|{{val|33.970564206}}

|-

|7

|{{val|1631432881}}

|{{val|40391}}

|{{val|57121}}

|align=left|{{val|1.41420118}}

|align=left|{{val|33.970562791}}

|-

|8

|{{val|55420693056}}

|{{val|235416}}

|{{val|332928}}

|align=left|{{val|1.41421144}}

|align=left|{{val|33.970562750}}

|-

|9

|{{val|1882672131025}}

|{{val|1372105}}

|{{val|1940449}}

|align=left|{{val|1.41421320}}

|align=left|{{val|33.970562749}}

|-

|10

|{{val|63955431761796}}

|{{val|7997214}}

|{{val|11309768}}

|align=left|{{val|1.41421350}}

|align=left|{{val|33.970562748}}

|-

|11

|{{val|2172602007770041}}

|{{val|46611179}}

|{{val|65918161}}

|align=left|{{val|1.41421355}}

|align=left|{{val|33.970562748}}

|}<!-- The table was generated in 23-jul-2016 using a Python script available at http://pastebin.com/sWyesrR8 -->

==See also==