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))

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.