All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers > Sequences
Double a triangle (Posted on 2015-02-19) Difficulty: 3 of 5
In the sequence of triangular numbers, some numbers are twice another.

For example t(20)=210 which is twice t(14)=105.

Characterize all such numbers.

Easy bonus: Explain why (except for the trivial case) there are no square numbers that are twice another.

No Solution Yet Submitted by Jer    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts One more observation Comment 3 of 3 |
Let x(x+1)/2 be the smaller number and y(y+1)/2 be the larger.  Then x(x+1) = y(y+1)/2.  This can be rearranged into (2x+1)^2 = (y+1)^2 + y^2.  

Then we can conclude these pairs of triangular numbers have a 1:1 correspondence to Pythagorean triangles whose legs are consecutive numbers.  Pythagorean triple (3,4,5) maps to triangular numbers t(2)=3 and t(3)=6; likewise (20,21,29) maps to t(14)=105 and t(20)=210 given in the problem statement.

  Posted by Brian Smith on 2016-07-30 11:24:32
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (5)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2018 by Animus Pactum Consulting. All rights reserved. Privacy Information