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