Square triangular number 36 depicted as a triangular number and as a square number.

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:

0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (sequence A001110 in the OEIS)

Explicit formulas

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 OEISA001110, OEISA001109, and OEISA001108 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 

Recurrence relations

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) 

We have[1][2]: 13 

Other characterizations

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]

See also


  1. ^ a b Dickson, Leonard Eugene (1999) [1920]. History of the Theory of Numbers. Vol. 2. Providence: American Mathematical Society. p. 16. ISBN 978-0-8218-1935-7.
  2. ^ a b c Euler, Leonhard (1813). "Regula facilis problemata Diophantea per numeros integros expedite resolvendi (An easy rule for Diophantine problems which are to be resolved quickly by integral numbers)". Mémoires de l'Académie des Sciences de St.-Pétersbourg (in Latin). 4: 3–17. Retrieved 2009-05-11. According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.
  3. ^ Weisstein, Eric W. "Square Triangular Number". MathWorld.
  4. ^ Ball, W. W. Rouse; Coxeter, H. S. M. (1987). Mathematical Recreations and Essays. New York: Dover Publications. p. 59. ISBN 978-0-486-25357-2.
  5. ^ Pietenpol, J. L.; Sylwester, A. V.; Just, Erwin; Warten, R. M. (February 1962). "Elementary Problems and Solutions: E 1473, Square Triangular Numbers". American Mathematical Monthly. 69 (2). Mathematical Association of America: 168–169. doi:10.2307/2312558. ISSN 0002-9890. JSTOR 2312558.
  6. ^ Plouffe, Simon (August 1992). "1031 Generating Functions" (PDF). University of Quebec, Laboratoire de combinatoire et d'informatique mathématique. p. A.129. Archived from the original (PDF) on 2012-08-20. Retrieved 2009-05-11.