Square triangular number

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For squares of triangular numbers, see squared 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).

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

$N_{k}=s_{k}^{2}={\frac {t_{k}(t_{k}+1)}{2}}.$

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

$N_{k}=\left({\frac {(3+2{\sqrt {2}})^{k}-(3-2{\sqrt {2}})^{k}}{4{\sqrt {2}}}}\right)^{2}.$

Other equivalent formulas (obtained by expanding this formula) that may be convenient include

${\begin{aligned}N_{k}&={1 \over 32}\left((1+{\sqrt {2}})^{{2k}}-(1-{\sqrt {2}})^{{2k}}\right)^{2}={1 \over 32}\left((1+{\sqrt {2}})^{{4k}}-2+(1-{\sqrt {2}})^{{4k}}\right)\\&={1 \over 32}\left((17+12{\sqrt {2}})^{k}-2+(17-12{\sqrt {2}})^{k}\right).\end{aligned}}$

The corresponding explicit formulas for s_k and t_k are ^[2]^:13

$s_{k}={\frac {(3+2{\sqrt {2}})^{k}-(3-2{\sqrt {2}})^{k}}{4{\sqrt {2}}}}$

and

$t_{k}={\frac {(3+2{\sqrt {2}})^{k}+(3-2{\sqrt {2}})^{k}-2}{4}}.$

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

${\frac {t(t+1)}{2}}=s^{2}.$

With a bit of algebra this becomes

$(2t+1)^{2}=8s^{2}+1,$

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

$x^{2}-2y^{2}=1$

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

$x=P_{{2k}}+P_{{2k-1}},\quad y=P_{{2k}};$

and therefore all solutions are given by

$s_{k}={\frac {P_{{2k}}}{2}},\quad t_{k}={\frac {P_{{2k}}+P_{{2k-1}}-1}{2}},\quad N_{k}=\left({\frac {P_{{2k}}}{2}}\right)^{2}.$

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)

$N_{k}=34N_{{k-1}}-N_{{k-2}}+2,{\text{ with }}N_{0}=0{\text{ and }}N_{1}=1.$

$N_{k}=\left(6{\sqrt {N_{{k-1}}}}-{\sqrt {N_{{k-2}}}}\right)^{2},{\text{ with }}N_{0}=1{\text{ and }}N_{1}=36.$

We have^[1]^[2]^:13

$s_{k}=6s_{{k-1}}-s_{{k-2}},{\text{ with }}s_{0}=0{\text{ and }}s_{1}=1;$

$t_{k}=6t_{{k-1}}-t_{{k-2}}+2,{\text{ with }}t_{0}=0{\text{ and }}t_{1}=1.$

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

${\frac {{\bigl (}4n(n+1){\bigr )}{\bigl (}4n(n+1)+1{\bigr )}}{2}}=2^{2}\,{\frac {n(n+1)}{2}}\,(2n+1)^{2}.$

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:^[8]

${\frac {1+z}{(1-z)(z^{2}-34z+1)}}=1+36z+1225z^{2}+\cdots .$

Numerical data

As k becomes larger, the ratio t_k / s_k approaches ${\sqrt {2}}\approx 1.41421$ and the ratio of successive square triangular numbers approaches $(1+{\sqrt {2}})^{4}=17+12{\sqrt {2}}\approx 33.97056$ .

${\begin{array}{rrrrll}k&N_{k}&s_{k}&t_{k}&t_{k}/s_{k}&N_{k}/N_{{k-1}}\\0&0&0&0&&\\1&1&1&1&1&\\2&36&6&8&1.33333&36\\3&1\,225&35&49&1.4&34.02778\\4&41\,616&204&288&1.41176&33.97224\\5&1\,413\,721&1\,189&1\,681&1.41379&33.97061\\6&48\,024\,900&6\,930&9\,800&1.41414&33.97056\\7&1\,631\,432\,881&40\,391&57\,121&1.41420&33.97056\\\end{array}}$

Notes

↑ 1.0 1.1 Dickson, Leonard Eugene (1999) [1920]. History of the Theory of Numbers 2. Providence: American Mathematical Society. p. 16. ISBN 978-0-8218-1935-7.
↑ 2.0 2.1 2.2 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. Cite uses deprecated parameters (help)
↑ 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.

External links

Triangular numbers that are also square at cut-the-knot
Weisstein, Eric W., "Square Triangular Number", MathWorld.
Michael Dummett's solution

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