Square triangular number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are an infinite number of square triangular numbers; the first few are 0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (sequence A001110 in OEIS).
Write Nk for the kth square triangular number, and write sk and tk for the sides of the corresponding square and triangle, so that
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
The corresponding explicit formulas for sk and tk are :13
With a bit of algebra this becomes
and then letting x = 2t + 1 and y = 2s, we get the Diophantine equation
and therefore all solutions are given by
There are many identities about the Pell numbers, and these translate into identities about the square triangular numbers.
A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers, to wit:
If the triangular number n(n+1)/2 is square, then so is the larger triangular number
We know this result has to be a square, because it is a product of three squares: 2^2 (by the exponent), (n(n+1))/2 (the n'th triangular number, by proof assumption), and the (2n+1)^2 (by the exponent). The product of any numbers that are squares is naturally going to result in another square, which can best be proven by geometrically visualizing the multiplication as the multiplying of a NxN box by an MxM box, which is done by placing one MxM box inside each cell of the NxN box, naturally producing another square result.
The generating function for the square triangular numbers is:
As k becomes larger, the ratio tk / sk approaches and the ratio of successive square triangular numbers approaches .
- Dickson, Leonard Eugene (1999) . History of the Theory of Numbers 2. Providence: American Mathematical Society. p. 16. ISBN 978-0-8218-1935-7.
- 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)". Memoires de l'academie des sciences de St.-Petersbourg (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."
- Barbeau, Edward (2003). Pell's Equation. Problem Books in Mathematics. New York: Springer. pp. 16–17. ISBN 978-0-387-95529-2. Retrieved 2009-05-10.
- Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford University Press. p. 210. ISBN 0-19-853171-0. "Theorem 244"
- Weisstein, Eric W., "Square Triangular Number", MathWorld.
- 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.
- Pietenpol, J. L.; A. V. Sylwester, Erwin Just, R. M Warten (February 1962). "Elementary Problems and Solutions: E 1473, Square Triangular Numbers". American Mathematical Monthly (Mathematical Association of America) 69 (2): 168–169. ISSN 0002-9890. JSTOR 2312558.
- Plouffe, Simon (August 1992). "1031 Generating Functions" (PDF). University of Quebec, Laboratoire de combinatoire et d'informatique mathématique. p. A.129. Retrieved 2009-05-11.