https://github.com/dvberkel/stn
A program to print out square triangular numbers.
https://github.com/dvberkel/stn
Last synced: 3 months ago
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A program to print out square triangular numbers.
- Host: GitHub
- URL: https://github.com/dvberkel/stn
- Owner: dvberkel
- License: mit
- Created: 2014-07-05T08:17:21.000Z (almost 11 years ago)
- Default Branch: master
- Last Pushed: 2014-07-05T09:49:58.000Z (almost 11 years ago)
- Last Synced: 2025-02-07T23:31:44.878Z (4 months ago)
- Language: Python
- Size: 33.6 MB
- Stars: 0
- Watchers: 3
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
Square Triangle Numbers
=======================
A program to print out square triangular numbers.
Running
-------
You can run the program with the following command
```sh
./stn.py [20]
```
the argument passed is the number of pairs that you want to output.
Background
----------
### Definition
We call *S(m)* the mth triangular number, i.e.
> S(m) = 1 + 2 + 3 + ... + m
By a [famous trick][trick] of [Gauss][], one has
> S(m) = m(m + 1) / 2
### Problem Statement
The problem asks for *m* and *n* such that
> S(m-1) = S(n) - S(m)
A little algebra reveals that this is equivalent to
> S(n) = S(m-1) + S(m) = m^2
The last identity becomes clear from a [proof without words][pww].
In other words, we are looking for a triangular number that is also a
square.
### Pell's Equation
From the formula for the nth triangular number
> S(n) = n(n + 1) / 2
We get, by using the identity *(a - b) (a + b) = a^2 - b^2*
> n (n + 1) / 2 = (n + 1/2 - 1/2) (n + 1/2 + 1/2) / 2 = [(2n + 1)^2 - 1] / 8
Equating this to the square *m^2*
> (2n + 1)^2 = 8m^2 - 1
Substituting *x = 2n + 1* and *y = 2m* this reduces to
> x^2 - 2y^2 = 1
which is an instance of [Pell's equation][pell]
### Solution
Pell's equation can be solved by means of finding convergents of
the [continued fraction][] expansion of the square root of two.
Using this method and substituting the solutions back into its
original form one gets the recurrence relations as stated below
> s(n) = 6s(n-1) - s(n-2)
> t(n) = 6t(n-1) - t(n-2) + 2
[trick]: http://mathcentral.uregina.ca/QQ/database/QQ.02.06/jo1.html
[Gauss]: http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
[pww]: http://www.cut-the-knot.org/ctk/pww.shtml
[pell]: http://en.wikipedia.org/wiki/Pell%27s_equation
[continued fraction]: http://en.wikipedia.org/wiki/Pell%27s_equation