Square 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:$$N_k = {1 over 32} left( left( 1 + sqrt{2} ight)^{2k} - left( 1 - sqrt{2} ight)^{2k} ight)^2 . or by the linear recursion:$$N_k = 34N_{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, ... OEIS|id=A001110

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

:$$(2n+1)^2=8m^2+1,

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

:$$k^2=2h^2+1

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

We get the recursion

:$$m_{k}=6m_{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} floor.

"t" is given by the formula:$$t(N_k) = {1 over 4} left [ left( left( 1 + sqrt{2} ight)^k + left( 1 - sqrt{2} ight)^k ight)^2 - left( 1 + (-1)^k ight)^2 ight] .

or by the recursion:$$t_k = 2sqrt{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))

$$egin{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 = 1.4142011end{matrix}

References

*cite journal
author = Sesskin, Sam
title = A "converse" to Fermat's last theorem?
journal = Mathematics Magazine
volume = 35
issue = 4
year = 1962
pages = 215–217
url = http://links.jstor.org/sici?sici=0025-570X(196209)35%3A4%3C215%3AA%22TFLT%3E2.0.CO%3B2-6

External links

* [http://www.cut-the-knot.org/do_you_know/triSquare.shtml Triangular numbers that are also square] at cut-the-knot
* [http://www.research.att.com/projects/OEIS?Anum=A001110 Sequence A001110] from the On-Line Encyclopedia of Integer Sequences.