# 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, 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

$N_{k}=s_{k}^{2}={\frac {t_{k}(t_{k}+1)}{2}}.$ Define the triangular root of a triangular number N = n(n + 1)/2 to be n. From this definition and the quadratic formula,

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

{\begin{aligned}x_{k+1}&=2x_{k}x_{1}-x_{k-1},\\y_{k+1}&=2y_{k}x_{1}-y_{k-1}.\end{aligned}} 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:

$s_{k}=y_{k},\quad t_{k}={\frac {x_{k}-1}{2}},\quad N_{k}=y_{k}^{2}.$ 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

$N_{k}=\left({\frac {\left(3+2{\sqrt {2}}\right)^{k}-\left(3-2{\sqrt {2}}\right)^{k}}{4{\sqrt {2}}}}\right)^{2}.$ Other equivalent formulas (obtained by expanding this formula) that may be convenient include

{\begin{aligned}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{aligned}} The corresponding explicit formulas for sk and tk are::13

{\begin{aligned}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{aligned}} ## 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

${\frac {t(t+1)}{2}}=s^{2}.$ Rearranging, this becomes

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

$x=P_{2k}+P_{2k-1},\quad y=P_{2k};$ 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:(12)

{\begin{aligned}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{aligned}} We have:13

{\begin{aligned}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{aligned}} ## Other characterizations

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

A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers, to wit:

If the nth triangular number n(n + 1)/2 is square, then so is the larger 4n(n + 1)th triangular number, since:

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

$N_{k}-N_{k-1}=s_{2k-1};$ 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:

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