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POHow does this regex find triangular numbers?
 

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  1. POHow does this regex find triangular numbers?
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    <blockquote> <p><sub><em>Part of a series of educational regex articles, this is a gentle introduction to the concept of nested references.</em></sub></p> </blockquote> <p>The first few <a href="http://en.wikipedia.org/wiki/Triangular_number" rel="nofollow noreferrer">triangular numbers</a> are:</p> <pre><code> 1 = 1 3 = 1 + 2 6 = 1 + 2 + 3 10 = 1 + 2 + 3 + 4 15 = 1 + 2 + 3 + 4 + 5 </code></pre> <p>There are many ways to check if a number is triangular. There's this interesting technique that uses regular expressions as follows:</p> <ul> <li>Given <em>n</em>, we first create a string of length <em>n</em> filled with the same character</li> <li>We then match this string against the pattern <code>^(\1.|^.)+$</code> <ul> <li><em>n</em> is triangular if and only if this pattern matches the string</li> </ul></li> </ul> <p>Here are some snippets to show that this works in several languages:</p> <h3><a href="http://ideone.com/KoS1a" rel="nofollow noreferrer">PHP (on ideone.com)</a></h3> <pre><code>$r = '/^(\1.|^.)+$/'; foreach (range(0,50) as $n) { if (preg_match($r, str_repeat('o', $n))) { print("$n "); } } </code></pre> <h3><a href="http://ideone.com/8FxQW" rel="nofollow noreferrer">Java (on ideone.com)</a></h3> <pre><code>for (int n = 0; n &lt;= 50; n++) { String s = new String(new char[n]); if (s.matches("(\\1.|^.)+")) { System.out.print(n + " "); } } </code></pre> <h3><a href="http://ideone.com/L61sT" rel="nofollow noreferrer">C# (on ideone.com)</a></h3> <pre><code>Regex r = new Regex(@"^(\1.|^.)+$"); for (int n = 0; n &lt;= 50; n++) { if (r.IsMatch("".PadLeft(n))) { Console.Write("{0} ", n); } } </code></pre> <p>So this regex seems to work, but can someone explain how?</p> <h3>Similar questions</h3> <ul> <li><a href="https://stackoverflow.com/questions/2795065/how-to-determine-if-a-number-is-a-prime-with-regex">How to determine if a number is a prime with regex?</a></li> </ul>
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    1. COThis series was started by permission of some in the community (http://meta.stackexchange.com/questions/62695/permission-to-start-a-series-of-advanced-regex-articles). If the reception is good, I plan to continue to cover other more advanced as well as more basic features of regex.
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    2. COIf this is supposed to be educational and for the community, why isn't it community wiki?
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    3. COI think the effort is worth some reputation. Please don't force them into community wiki. Who cares if poly goes from 44k to 50k, what's the difference?
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