Square triangular number

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For squares of triangular numbers, see squared triangular number.

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 triangular squares, given by the formula

$N_k = {1 \over 32} \left( \left( 1 + \sqrt{2} \right)^{2k} - \left( 1 - \sqrt{2} \right)^{2k} \right)^2 .$

or by the linear recursion

N k = 34 N k - 1 - N k - 2 + 2

with

N 0 = 0

and

N 1 = 1

The first few square triangular numbers are 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, ... (sequence A001110 in OEIS)

The problem of finding square triangular numbers reduces to Pell's equation in the following way. Every triangular number is of the form n(n + 1)/2. Therefore we seek integers n, m such that

n (n + 1) / 2 = m 2 .

With a bit of algebra this becomes

(2 n + 1) 2 = 8 m 2 + 1,

and then letting k = 2n + 1 and h = 2m, we get the Diophantine equation

k 2 = 2 h 2 + 1

which is an instance of Pell's equation and is solved by the Pell numbers.

We get the recursion

m k = 6 m k - 1 - m k - 2 .

Also, note that

$m^2_{k}-1=m_{k+1}m_{k-1}$

since $m 0 = 1$ and $m 1 = 6$ .

The k^th triangular square N_k is equal to the s^th perfect square and the t^th triangular number, such that

$s(N) = \sqrt{N},$

$t(N) = \lfloor \sqrt{2 N} \rfloor.$

t is given by the formula

$t(N_k) = {1 \over 4} \left[ \left( \left( 1 + \sqrt{2} \right)^k + \left( 1 - \sqrt{2} \right)^k \right)^2 - \left( 1 + (-1)^k \right)^2 \right].$

or by the recursion

$t_k = 2\sqrt{2t_{k-1}(t_{k-1}+1)} + 3t_{k-1} + 1$

As k becomes larger, the ratio t/s approaches the square root of two: Also ratio of successive square triangulars converges to 17+12(sqrt(2))

$\begin{matrix} N=1 & s=1 & t=1 & t/s=1 \\ N=36 & s=6 & t=8 & t/s = 1.3333333 \\ N=1225 & s=35 & t=49 & t/s = 1.4 \\ N=41616 & s=204 & t=288 & t/s = 1.4117647 \\ N=1,413,721 & s=1189 & t=1681 & t/s = 1.4137931 \\ N=48,024,900 & s=6930 & t=9800 & t/s = 1.4141414 \\ N=1,631,432,881 & s=40391 & t=57121 & t/s = 1.4142011 \end{matrix}$