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Square triangular number: Wikis

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

Contents

Explicit formulas

Write Nk for the kth square triangular number, and write sk and tk 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 Nk, sk and tk 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{align} 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{align}

The corresponding explicit formulas for sk and tk 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 = 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 Pk 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)

Nk = 34Nk − 1Nk − 2 + 2, with N0 = 0 and N1 = 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

sk = 6sk − 1sk − 2, with s0 = 0 and s1 = 1;
tk = 6tk − 1tk − 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

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

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

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

Numerical data

As k becomes larger, the ratio tk / sk approaches \sqrt{2} \approx 1.41421 and the ratio of successive square triangular numbers approaches 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. ^ 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/books?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/books?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 69 (2): 168–169. ISSN 00029890. http://www.jstor.org/stable/2312558. Retrieved 2009-05-11. 
  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. 

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