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 A triangular number is a number that can be arranged in the shape of an equilateral triangle. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer. ... The word infinity comes from the Latin infinitas or unboundedness. It refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ...

or by the linear recursion A Sierpinski triangle —a confined recursion of triangles to form a geometric lattice. ...

N_k = 34N_{k − 1} − N_{k − 2} + 2 with N₀ = 0 and N₁ = 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 Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

n(n + 1) / 2 = m².

With a bit of algebra this becomes

(2n + 1)² = 8m² + 1,

and then letting k = 2n + 1 and h = 2m, we get the Diophantine equation In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ...

k² = 2h² + 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

t is given by the formula

As k becomes larger, the ratio t/s approaches the square root of two: Also ratio of successive square triangulars converges to 17+12(sqrt(2))

External references

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