- 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:or by the linearrecursion : with andThe 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:
With a bit of algebra this becomes
:
and then letting "k" = 2"n" + 1 and "h" = 2"m", we get the
Diophantine equation :
which is an instance of Pell's equation and is solved by the
Pell number s.We get the
recursion :
Also, note that
:
since and .
The "kth" triangular square "Nk" is equal to the "sth" perfect square and the "tth" triangular number, such that::
"t" is given by the formula:
or by the recursion:
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%3B2-6External 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 theOn-Line Encyclopedia of Integer Sequences .
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