Triangular square number
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A triangular square number is a number which is both a triangular number and a perfect square. There is an infinity of triangular squares, given by the formula
- <math> N_k = {1 \over 32} \left( \left( 1 + \sqrt{2} \right)^{2k} - \left( 1 - \sqrt{2} \right)^{2k} \right)^2 . <math>
The problem of finding triangular square numbers reduces to Pell's equation in the following way. Every triangular number is of the form n(n − 1)/2. Therefore we seek integers n, m such that
- <math>n(n+1)/2 = m^2.<math>
With a bit of algebra this becomes
- <math>(2n+1)^2=8m^2+1,<math>
and then letting k = 2n + 1 and h = 2m, we get the Diophantine equation
- <math>k^2=2h^2+1<math>
which is an instance of Pell's equation.
The kth triangular square Nk is equal to the sth perfect square and the tth triangular number, such that
- <math> s(N) = \sqrt{N}, <math>
- <math> t(N) = \lfloor \sqrt{2 N} \rfloor. <math>
t is given by the formula
- <math> t(N_k) = {1 \over 4} \left[ \left( \left( 1 + \sqrt{2} \right)^k + \left( 1 - \sqrt{2} \right)^k \right)^2 - \left( 1 + (-1)^k \right)^2 \right] <math>.
As k becomes larger, the ratio t/s approaches the square root of two:
<math> \begin{matrix} N=1 & s=1 & t=1 & t/s=1 \\ N=36 & s=6 & t=8 & t/s = 1.3333333 \\ N=1225 & s=35 & t=49 & t/s = 1.4 \\ N=41616 & s=204 & t=288 & t/s = 1.4117647 \\ N=1,413,721 & s=1189 & t=1681 & t/s = 1.4137931 \\ N=48,024,900 & s=6930 & t=9800 & t/s = 1.4141414 \\ N=1,631,432,881 & s=40391 & t=57121 & t/s = 1.4142011 \end{matrix} <math>
External references
- Triangular numbers that are also square (http://www.cut-the-knot.org/do_you_know/triSquare.shtml). From Interactive Mathematics Miscellany and Puzzles.fr:Nombre carré triangulaire
it:Numero quadrato triangolare sl:trikotniško kvadratno število zh:三角平方數