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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    
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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
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