In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number. There are infinitely many square triangular numbers; the first few are:
Write for the th square triangular number, and write and for the sides of the corresponding square and triangle, so that
Define the triangular root of a triangular number to be . From this definition and the quadratic formula,
Therefore, is triangular ( is an integer) if and only if is square. Consequently, a square number is also triangular if and only if is square, that is, there are numbers and such that . This is an instance of the Pell equation with . All Pell equations have the trivial solution for any ; this is called the zeroth solution, and indexed as . If denotes the th nontrivial solution to any Pell equation for a particular , it can be shown by the method of descent that the next solution is
Hence there are infinitely many solutions to any Pell equation for which there is one non-trivial one, which is true whenever is not a square. The first non-trivial solution when is easy to find: it is . A solution to the Pell equation for yields a square triangular number and its square and triangular roots as follows:
Hence, the first square triangular number, derived from , is , and the next, derived from , is .
The sequences , and are the OEIS sequences OEIS: A001110, OEIS: A001109, and OEIS: A001108 respectively.
In 1778 Leonhard Euler determined the explicit formula [1] [2]: 12–13
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
The corresponding explicit formulas for and are: [2]: 13
There are recurrence relations for the square triangular numbers, as well as for the sides of the square and triangle involved. We have [3]: (12)
All square triangular numbers have the form , where is a convergent to the continued fraction expansion of , the square root of 2. [4]
A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the th triangular number is square, then so is the larger 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. [5]
The generating function for the square triangular numbers is: [6]
According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number. There are infinitely many square triangular numbers; the first few are:
Write for the th square triangular number, and write and for the sides of the corresponding square and triangle, so that
Define the triangular root of a triangular number to be . From this definition and the quadratic formula,
Therefore, is triangular ( is an integer) if and only if is square. Consequently, a square number is also triangular if and only if is square, that is, there are numbers and such that . This is an instance of the Pell equation with . All Pell equations have the trivial solution for any ; this is called the zeroth solution, and indexed as . If denotes the th nontrivial solution to any Pell equation for a particular , it can be shown by the method of descent that the next solution is
Hence there are infinitely many solutions to any Pell equation for which there is one non-trivial one, which is true whenever is not a square. The first non-trivial solution when is easy to find: it is . A solution to the Pell equation for yields a square triangular number and its square and triangular roots as follows:
Hence, the first square triangular number, derived from , is , and the next, derived from , is .
The sequences , and are the OEIS sequences OEIS: A001110, OEIS: A001109, and OEIS: A001108 respectively.
In 1778 Leonhard Euler determined the explicit formula [1] [2]: 12–13
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
The corresponding explicit formulas for and are: [2]: 13
There are recurrence relations for the square triangular numbers, as well as for the sides of the square and triangle involved. We have [3]: (12)
All square triangular numbers have the form , where is a convergent to the continued fraction expansion of , the square root of 2. [4]
A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the th triangular number is square, then so is the larger 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. [5]
The generating function for the square triangular numbers is: [6]
According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.