A square triangular number (or 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

$N\_k\; =\; \{1\; \backslash over\; 32\}\; \backslash left(\; \backslash left(\; 1\; +\; \backslash sqrt\{2\}\; \backslash right)^\{2k\}\; -\; \backslash left(\; 1\; -\; \backslash sqrt\{2\}\; \backslash right)^\{2k\}\; \backslash right)^2\; .$

$N\_k\; =\; 34N\_\{k-1\}\; -\; N\_\{k-2\}\; +\; 2$ with $N\_0\; =\; 0$ and $N\_1\; =\; 1$

The problem of finding square triangular 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

$n(n+1)/2\; =\; m^2.$

With a bit of algebra this becomes

$(2n+1)^2=8m^2+1,$

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

$k^2=2h^2+1$

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

$s(N)\; =\; \backslash sqrt\{N\},$

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

t is given by the formula

$t(N\_k)\; =\; \{1\; \backslash over\; 4\}\; \backslash left\backslash left(\; \backslash left(\; 1\; +\; \backslash sqrt\{2\}\; \backslash right)^k\; +\; \backslash left(\; 1\; -\; \backslash sqrt\{2\}\; \backslash right)^k\; \backslash right)^2\; -\; \backslash left(\; 1\; +\; (-1)^k\; \backslash right)^2\; \backslash right.$

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

External references

• Triangular numbers that are also square at cut-the-knot
• Sequence A001110 from the On-Line Encyclopedia of Integer Sequences.

Figurate numbers

Nombre carré triangulaire | Numero quadrato triangolare | מספר משולשי ריבועי | Trikotniško kvadratno število | 三角平方數