In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number has a square root that is an integer. There are infinitely many square triangular numbers; the first few are:

Solution as a Pell equation

for the sides of the corresponding square and triangle, so that

Define the triangular root. From the quadratic formula,

is an integer) if and only if. This is an instance of the Pell equation, it can be shown by the method of descent that the next solution is yields a square triangular number and its square and triangular roots as follows:

.

are the OEIS sequences , , and respectively.

Explicit formula

In 1778 Leonhard Euler determined the explicit formula

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

are:

Recurrence relations

The solution to the Pell equation can be expressed as a recurrence relation for the equation's solutions. This can be translated into recurrence equations that directly express the square triangular numbers, as well as the sides of the square and triangle involved. We have

We have

Other characterizations

is a convergent to the continued fraction expansion, the square root of 2.

th triangular number, since:

The left hand side of this equation is in the form of a triangular number, and as the product of three squares, the right hand side is square.

The generating function for the square triangular numbers is:

See also

• Cannonball problem, on numbers that are simultaneously square and square pyramidal

• Sixth power, numbers that are simultaneously square and cubical

Notes

External links

• Triangular numbers that are also square at cut-the-knot

• Michael Dummett's solution