Editing Square triangular number (section)

==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 }}
</ref><ref name=Euler>
{{cite journal |last=Euler |first=Leonhard |author-link=Leonhard Euler |year=1813 |title=Regula facilis problemata Diophantea per numeros integros expedite resolvendi (An easy rule for Diophantine problems which are to be resolved quickly by integral numbers)  |journal=Mémoires de l'Académie des Sciences de St.-Pétersbourg |volume= 4 |pages=3–17 |url=http://math.dartmouth.edu/~euler/pages/E739.html |language=la |access-date=2009-05-11 |quote=According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.}}
</ref>{{Rp|12–13}}

{{bi|left=1.6|<math>\displaystyle N_k = \left( \frac{\left(3 + 2\sqrt{2}\right)^k - \left(3 - 2\sqrt{2}\right)^k}{4\sqrt{2}} \right)^2.
</math>}}

Other equivalent formulas (obtained by expanding this formula) that may be convenient include

{{bi|left=1.6|<math>\displaystyle \begin{align} 
N_k &= \tfrac{1}{32} \left( \left( 1 + \sqrt{2} \right)^{2k} - \left( 1 - \sqrt{2} \right)^{2k} \right)^2 \\
&= \tfrac{1}{32} \left( \left( 1 + \sqrt{2} \right)^{4k}-2 + \left( 1 - \sqrt{2} \right)^{4k} \right) \\
&= \tfrac{1}{32} \left( \left( 17 + 12\sqrt{2} \right)^k -2 + \left( 17 - 12\sqrt{2} \right)^k \right).
\end{align}</math>}}

The corresponding explicit formulas for <math>s_k</math> and <math>t_k</math> are:<ref name=Euler />{{Rp|13}}

{{bi|left=1.6|<math>\displaystyle \begin{align}
s_k &= \frac{\left(3 + 2\sqrt{2}\right)^k - \left(3 - 2\sqrt{2}\right)^k}{4\sqrt{2}}, \\
t_k &= \frac{\left(3 + 2\sqrt{2}\right)^k + \left(3 - 2\sqrt{2}\right)^k - 2}{4}.
\end{align}</math>}}