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 infinitely many square triangular numbers; the first few are:
 0, 1, 36, 225, 1616, 41413721, 1024900, 48631432881, 1420693056, 55882672131025 (sequence 1A001110 in the OEIS)
Contents
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
Define the triangular root of a triangular number N = n(n + 1)/2 to be n. From this definition and the quadratic formula,
Therefore, N is triangular (n is an integer) if and only if 8N + 1 is square. Consequently, a square number M^{2} is also triangular if and only if 8M^{2} + 1 is square, that is, there are numbers x and y such that x^{2} − 8y^{2} = 1. This is an instance of the Pell equation with n = 8. All Pell equations have the trivial solution x = 1, y = 0 for any n; this is called the zeroth solution, and indexed as (x_{0}, y_{0}) = (1,0). If (x_{k}, y_{k}) denotes the kth nontrivial solution to any Pell equation for a particular n, it can be shown by the method of descent that
Hence there are an infinity of solutions to any Pell equation for which there is one nontrivial one, which holds whenever n is not a square. The first nontrivial solution when n = 8 is easy to find: it is (3,1). A solution (x_{k}, y_{k}) to the Pell equation for n = 8 yields a square triangular number and its square and triangular roots as follows:
Hence, the first square triangular number, derived from (3,1), is 1, and the next, derived from 6 × (3,1) − (1,0) = (17,6), is 36.
The sequences N_{k}, s_{k} and t_{k} 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 s_{k} and t_{k} are:^{[2]}^{:13}
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
Rearranging, this becomes
and then letting x = 2t + 1 and y = 2s, we get the Diophantine equation
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)}
We have^{[1]}^{[2]}^{:13}
Other characterizations
All square triangular numbers have the form b^{2}c^{2}, where b/c is a convergent to the continued fraction for the √2.^{[6]}
A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers, to wit:^{[7]}
If the nth triangular number n(n + 1)/2 is square, then so is the larger 4n(n + 1)th triangular number, since:
We know this result has to be a square, because it is a product of three squares: 4, n(n + 1)/2 (the original square triangular number), and (2n + 1)^{2}.
The triangular roots t_{k} are alternately simultaneously one less than a square and twice a square if k is even, and simultaneously a square and one less than twice a square if k is odd. Thus,
 49 = 7^{2} = 2 × 5^{2} − 1,
 288 = 17^{2} − 1 = 2 × 12^{2}, and
 1681 = 41^{2} = 2 × 29^{2} − 1.
In each case, the two square roots involved multiply to give s_{k}: 5 × 7 = 35, 12 × 17 = 204, and 29 × 41 = 1189.^{[citation needed]}
Additionally:
36 − 1 = 35, 1225 − 36 = 1189, and 41616 − 1225 = 40391. In other words, the difference between two consecutive square triangular numbers is the square root of another square triangular number.^{[citation needed]}
The generating function for the square triangular numbers is:^{[8]}
Numerical data
As k becomes larger, the ratio t_{k}/s_{k} approaches √2 ≈ 21356, and the ratio of successive square triangular numbers approaches 1.414(1 + √2)^{4} = 17 + 12√2 ≈ 562748. The table below shows values of 33.970k between 0 and 11, which comprehend all square triangular numbers up to . 10^{16}
k N_{k} s_{k} t_{k} t_{k}/s_{k} N_{k}/N_{k − 1} 0 0 0 0 1 1 1 1 1 2 36 6 8 33333 1.333 36 3 225 1 35 49 1.4 777778 34.027 4 616 41 204 288 76471 1.411 244898 33.972 5 413721 1 189 1 681 1 79310 1.413 612265 33.970 6 024900 48 930 6 800 9 14141 1.414 564206 33.970 7 631432881 1 391 40 121 57 20118 1.414 562791 33.970 8 420693056 55 416 235 928 332 21144 1.414 562750 33.970 9 882672131025 1 372105 1 940449 1 21320 1.414 562749 33.970 10 955431761796 63 997214 7 309768 11 21350 1.414 562748 33.970 11 172602007770041 2 611179 46 918161 65 21355 1.414 562748 33.970
See also
 Cannonball problem, on numbers that are simultaneously square and square pyramidal
 Sixth power, numbers that are simultaneously square and cubical
Notes
 ^ ^{a} ^{b} Dickson, Leonard Eugene (1999) [1920]. History of the Theory of Numbers. 2. Providence: American Mathematical Society. p. 16. ISBN 9780821819357.

^ ^{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 20090511.
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 9780387955292. Retrieved 20090510.

^
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
 ^ 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 9780486253572.
 ^ 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. Mathematical Association of America. 69 (2): 168–169. doi:10.2307/2312558. ISSN 00029890. JSTOR 2312558.
 ^ Plouffe, Simon (August 1992). "1031 Generating Functions" (PDF). University of Quebec, Laboratoire de combinatoire et d'informatique mathématique. p. A.129. Retrieved 20090511.
External links
 Triangular numbers that are also square at cuttheknot
 Weisstein, Eric W. "Square Triangular Number". MathWorld.
 Michael Dummett's solution