Square triangular number

A triangular square number (or square triangular 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

${\displaystyle N_{k}={1 \over 32}\left(\left(1+{\sqrt {2}}\right)^{2k}-\left(1-{\sqrt {2}}\right)^{2k}\right)^{2}.}$

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

${\displaystyle n(n+1)/2=m^{2}.}$

With a bit of algebra this becomes

${\displaystyle (2n+1)^{2}=8m^{2}+1,}$

and then letting k = 2n + 1 and h = 2m, we get the Diophantine equation

${\displaystyle k^{2}=2h^{2}+1}$

which is an instance of Pell's equation.

The k^th triangular square N_k is equal to the s^th perfect square and the t^th triangular number, such that

${\displaystyle s(N)={\sqrt {N}},}$

${\displaystyle t(N)=\lfloor {\sqrt {2N}}\rfloor .}$

t is given by the formula

${\displaystyle t(N_{k})={1 \over 4}\left[\left(\left(1+{\sqrt {2}}\right)^{k}+\left(1-{\sqrt {2}}\right)^{k}\right)^{2}-\left(1+(-1)^{k}\right)^{2}\right]}$.

As k becomes larger, the ratio t/s approaches the square root of two:

${\displaystyle {\begin{matrix}N=1&s=1&t=1&t/s=1\\N=36&s=6&t=8&t/s=1.3333333\\N=1225&s=35&t=49&t/s=1.4\\N=41616&s=204&t=288&t/s=1.4117647\\N=1,413,721&s=1189&t=1681&t/s=1.4137931\\N=48,024,900&s=6930&t=9800&t/s=1.4141414\\N=1,631,432,881&s=40391&t=57121&t/s=1.4142011\end{matrix}}}$

External references

• Triangular numbers that are also square. From Interactive Mathematics Miscellany and Puzzles.