# Square triangular number

Short description: Integer that is both a perfect square and a 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, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (sequence A001110 in the OEIS)

## 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

$\displaystyle{ N_k = s_k^2 = \frac{t_k(t_k+1)}{2}. }$

Define the triangular root of a triangular number N = n(n + 1)/2 to be n. From this definition and the quadratic formula,

$\displaystyle{ n = \frac{\sqrt{8N + 1} - 1}{2}. }$

Therefore, N is triangular (n is an integer) if and only if 8N + 1 is square. Consequently, a square number M2 is also triangular if and only if 8M2 + 1 is square, that is, there are numbers x and y such that x2 − 8y2 = 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 (x0, y0) = (1,0). If (xk, yk) denotes the kth nontrivial solution to any Pell equation for a particular n, it can be shown by the method of descent that

\displaystyle{ \begin{align} x_{k+1} &= 2x_k x_1 - x_{k-1}, \\ y_{k+1} &= 2y_k x_1 - y_{k-1}. \end{align} }

Hence there are an infinity of solutions to any Pell equation for which there is one non-trivial one, which holds whenever n is not a square. The first non-trivial solution when n = 8 is easy to find: it is (3,1). A solution (xk, yk) to the Pell equation for n = 8 yields a square triangular number and its square and triangular roots as follows:

$\displaystyle{ s_k = y_k , \quad t_k = \frac{x_k - 1}{2}, \quad N_k = y_k^2. }$

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 Nk, sk and tk are the OEIS sequences , , and respectively.

In 1778 Leonhard Euler determined the explicit formula:12–13

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

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

\displaystyle{ \begin{align} N_k &= \tfrac{1}{32} \left( \left( 1 + \sqrt{2} \right)^{2k} - \left( 1 - \sqrt{2} \right)^{2k} \right)^2 \\ &= \tfrac{1}{32} \left( \left( 1 + \sqrt{2} \right)^{4k}-2 + \left( 1 - \sqrt{2} \right)^{4k} \right) \\ &= \tfrac{1}{32} \left( \left( 17 + 12\sqrt{2} \right)^k -2 + \left( 17 - 12\sqrt{2} \right)^k \right). \end{align} }

The corresponding explicit formulas for sk and tk are::13

\displaystyle{ \begin{align} s_k &= \frac{\left(3 + 2\sqrt{2}\right)^k - \left(3 - 2\sqrt{2}\right)^k}{4\sqrt{2}}, \\ t_k &= \frac{\left(3 + 2\sqrt{2}\right)^k + \left(3 - 2\sqrt{2}\right)^k - 2}{4}. \end{align} }

## Pell's equation

The problem of finding square triangular numbers reduces to Pell's equation in the following way.

Every triangular number is of the form t(t + 1)/2. Therefore we seek integers t, s such that

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

Rearranging, this becomes

$\displaystyle{ \left(2t+1\right)^2=8s^2+1, }$

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

$\displaystyle{ x^2 - 2y^2 =1, }$

which is an instance of Pell's equation. This particular equation is solved by the Pell numbers Pk as

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

and therefore all solutions are given by

$\displaystyle{ s_k = \frac{P_{2k}}{2}, \quad t_k = \frac{P_{2k} + P_{2k-1} -1}{2}, \quad N_k = \left( \frac{P_{2k}}{2} \right)^2. }$

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:(12)

\displaystyle{ \begin{align} 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 &= 0\text{ and }N_1 = 1. \end{align} }

We have:13

\displaystyle{ \begin{align} 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. \end{align} }

## Other characterizations

All square triangular numbers have the form b2c2, where b/c is a convergent to the continued fraction expansion of 2.

A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers: If the nth triangular number n(n + 1)/2 is square, then so is the larger 4n(n + 1)th triangular number, since:

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

As the product of three squares, the right hand side is square. The triangular roots tk 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 = 72 = 2 × 52 − 1,
288 = 172 − 1 = 2 × 122, and
1681 = 412 = 2 × 292 − 1.

In each case, the two square roots involved multiply to give sk: 5 × 7 = 35, 12 × 17 = 204, and 29 × 41 = 1189.

$\displaystyle{ N_k - N_{k-1}=s_{2k-1}; }$

The generating function for the square triangular numbers is:

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

## Numerical data

As k becomes larger, the ratio tk/sk approaches 2 ≈ 1.41421356, and the ratio of successive square triangular numbers approaches (1 + 2)4 = 17 + 122 ≈ 33.970562748. The table below shows values of k between 0 and 11, which comprehend all square triangular numbers up to 1016.

k Nk sk tk tk/sk Nk/Nk − 1
0 0 0 0
1 1 1 1 1
2 36 6 8 1.33333333 36
3 1225 35 49 1.4 34.027777778
4 41616 204 288 1.41176471 33.972244898
5 1413721 1189 1681 1.41379310 33.970612265
6 48024900 6930 9800 1.41414141 33.970564206
7 1631432881 40391 57121 1.41420118 33.970562791
8 55420693056 235416 332928 1.41421144 33.970562750
9 1882672131025 1372105 1940449 1.41421320 33.970562749
10 63955431761796 7997214 11309768 1.41421350 33.970562748
11 2172602007770041 46611179 65918161 1.41421355 33.970562748