Square Triangular Number Information

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).

Explicit formulas

Write N_k for the kth square triangular number, and write s_k and t_k for the sides of the corresponding square and triangle, so that

The sequences N_k, s_k and t_k are the OEIS sequences  A001110,  A001109, and  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 s_k and t_k are ^[2]:13

and

Pell's equation

The problem of finding square triangular numbers reduces to Pell's equation in the following way.^[3] Every triangular number is of the form t(t + 1)/2. Therefore we seek integers t, s such that

With a bit of algebra this becomes

(2t + 1)² = 8s² + 1,

and then letting x = 2t + 1 and y = 2s, we get the Diophantine equation

x² − 2y² = 1

which is an instance of Pell's equation. This particular equation is solved by the Pell numbers P_k as^[4]

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.

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^[5]:(12)

N_k = 34N_{k − 1} − N_{k − 2} + 2, with N₀ = 0 and N₁ = 1.

We have^[1][2]:13

s_k = 6s_{k − 1} − s_{k − 2}, with s₀ = 0 and s₁ = 1;

t_k = 6t_{k − 1} − t_{k − 2} + 2, with t₀ = 0 and t₁ = 1.

Other characterizations

All square triangular numbers have the form b²c², where b / c is a convergent to the continued fraction for the square root of 2.^[6]

A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers, to wit:^[7] If the triangular number n(n+1)/2 is square, then so is the larger triangular number

The generating function for the square triangular numbers is:^[8]

Numerical data

As k becomes larger, the ratio t_k / s_k approaches and the ratio of successive square triangular numbers approaches .

Notes

1. ^ ^{a b} Dickson, Leonard Eugene (1999) [1920]. History of the Theory of Numbers. 2. Providence: American Mathematical Society. p. 16. ISBN 9780821819357. http://books.google.com/?id=eNjKEBLt_tQC&pg=PA16. Retrieved 2009-05-10.
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)" (in Latin). Memoires de l'academie des sciences de St.-Petersbourg 4: 3–17. http://math.dartmouth.edu/~euler/pages/E739.html. Retrieved 2009-05-11. "According to the records, it was presented to the St. Petersburg Academy on May 4, 1778."
3. ^ Barbeau, Edward (2003). Pell's Equation. Problem Books in Mathematics. New York: Springer. pp. 16–17. ISBN 9780387955292. http://books.google.com/?id=FtoFImV5BKMC&pg=PA16. Retrieved 2009-05-10.
4. ^ Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford University Press. p. 210. ISBN 0198531710. "Theorem 244"
5. ^ Weisstein, Eric W., "Square Triangular Number" from MathWorld.
6. ^ Ball, W. W. Rouse; Coxeter, H. S. M. (1987). Mathematical Recreations and Essays. New York: Dover Publications. p. 59. ISBN 9780486253572.
7. ^ 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 00029890. JSTOR 2312558.
8. ^ Plouffe, Simon (August 1992). "1031 Generating Functions" (PDF). University of Quebec, Laboratoire de combinatoire et d'informatique mathématique. p. A.129. http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf. Retrieved 2009-05-11.

External links

• Triangular numbers that are also square at cut-the-knot
• Weisstein, Eric W., "Square Triangular Number" from MathWorld.

Categories:

• Figurate numbers
• Integer sequences

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See Also:

• Root Mean Square
• Square Wave