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Best way to determine if number is triangular number?

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I would like to test if an integer is a triangular number, and am wondering what the best way to achieve this is. I am aware that a triangular number is given by
x = n(n+1)/2
And that x is triangular if and only if 8x + 1 is a square.
However, what is the best way to actually test this in Matlab, giving consideration to floating point arithmetic?
I can think of methods like:
check x is an integer find if sqrt(8x + 1) is an integer to some floating point error
but there must be better ways.


John D'Errico
John D'Errico on 6 Sep 2012
Note that as long as 8x+1 is no more than 2^53-1, the simple test for a perfect square will be valid. And larger than that you cannot represent x exactly anyway as a double.
Richard Crozier
Richard Crozier on 6 Sep 2012
I will only need this for relatively small numbers, it's actually a test of some user input. Useful to bear in mind though, thanks.

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Accepted Answer

Matt Fig
Matt Fig on 6 Sep 2012
istri = @(x) floor(sqrt(8*x+1))==sqrt(8*x+1);
x = 1:20;
idx = istri(x)

  1 Comment

Richard Crozier
Richard Crozier on 6 Sep 2012
I think this is the best for my purposes, thanks!

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More Answers (2)

Azzi Abdelmalek
Azzi Abdelmalek on 6 Sep 2012
Edited: Azzi Abdelmalek on 6 Sep 2012
for n=1:100 %for example
x = n*(n+1)/2;
if round(y)==y
n=size(result) % you will obtain n=100, which verify the formula
n=1:100 %for example
x = n.*(n+1)/2;y=sqrt(8*x+1)
all(y==round(y)) % if the result is 1, the formula is checked


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Sean de Wolski
Sean de Wolski on 6 Sep 2012


Richard Crozier
Richard Crozier on 6 Sep 2012
This suggestion is fine, except that I can't actually see the solutions.
Sean de Wolski
Sean de Wolski on 6 Sep 2012
Sure you can! You just have to get addicted to it and solve a bunch of other problems first!

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