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.

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.