Help with numbers

[Shadow]

Hey all,

 

first, apologies for the non-descriptive title, but I really couldn't come up with anything better. That said, I'll get straight to the point; we have the following numbers:

 

[math]N_{1}=1[/math]

 

[math]N_{2}=3[/math]

 

[math]N_{3}=6[/math]

 

[math]N_{4}=10[/math]

 

[math]N_{5}=15[/math]

 

[math]N_{6}=21[/math]

 

[math]...[/math]

 

[math]N_{n}=N_{n-1}+n[/math]

 

Let me just give those numbers a name, for the sake of simplicity I'll call them Strange numbers (note, I wouldn't be surprised at all if this was a long discovered and frequently used series, such as the Fibonacci Numbers...so if it is, please correct me, since I am not aware of it's existence).

 

Now, my question; given the number [math]x[/math], could anyone tell me if there is any way to find out if the number [math]x[/math] is a "Strange number"? Some mathematical formula or some such rule for me to check against...ideally, although I'm pretty sure it's not going to be that easy, I'm looking for something along the lines of "A number is a square only if it's square root is an integer". I know that's not a formal definition, but it would probably be the easiest way to check in say a program, which is what I need this for in the first place.

 

My thanks to anyone willing to help,

 

Shadow

[doG]

Your sequence looks like the third column of Pascal's Triangle...

[Country Boy]

Hey all,

 

first, apologies for the non-descriptive title, but I really couldn't come up with anything better. That said, I'll get straight to the point; we have the following numbers:

 

[math]N_{1}=1[/math]

 

[math]N_{2}=3[/math]

 

[math]N_{3}=6[/math]

 

[math]N_{4}=10[/math]

 

[math]N_{5}=15[/math]

 

[math]N_{6}=21[/math]

 

[math]...[/math]

 

[math]N_{n}=N_{n-1}+n[/math]

 

Let me just give those numbers a name, for the sake of simplicity I'll call them Strange numbers (note, I wouldn't be surprised at all if this was a long discovered and frequently used series, such as the Fibonacci Numbers...so if it is, please correct me, since I am not aware of it's existence).

 

Now, my question; given the number [math]x[/math], could anyone tell me if there is any way to find out if the number [math]x[/math] is a "Strange number"? Some mathematical formula or some such rule for me to check against...ideally, although I'm pretty sure it's not going to be that easy, I'm looking for something along the lines of "A number is a square only if it's square root is an integer". I know that's not a formal definition, but it would probably be the easiest way to check in say a program, which is what I need this for in the first place.

 

My thanks to anyone willing to help,

 

Shadow

Those are triangular numbers. n is a triangular number if n objects can be placed in an equilateral triangle form. 3 objects is obvious. 6 objects form a triangle with rows containing 3, 2, and 1. 10 objects would be 4, 3, 2, 1 and is the standard form for bowling "ten-pens". If n objects are placed in a triangle in which the longest row contains k objects, to make a larger triangle, you need to add a row of k+1 objects. That way, N_(n+1)= N_n+ n.

 

A number is triangular if and only if it is of the form n(n+1)/2 for some positive integer n.

 

That is because the triangular numbers are those that can be written 1+ 2+ 3+ 4+ ...+ n. If you reverse the order and write n+ ...+ 4+ 3+ 2+ 1 and add first numbers of each sum, second numbers of each sum, etc., you will find that every sum is n+1 (top number increases by 1, bottom number decreases by 1 so the sum is always the same). Since there are n such sums, that total will be n(n+1). Since we got that by doing the sum twice the value of just one sum is n(n+1)/2.

[Shadow]

Wow. Thanks for the feedback ) Real apreciated