1+2-3=0; 1+2+3+...+12+13+14-15-16-...-19-20=0; why does this only occur for numbers (3,20,119,696...) which are leg-leg twin pythagorean triples?

Question

The pivot point (+13+14-15) (+83+84-85) tends to the 1/√2 of the sum. So 14/20, then 84/119, then 492/696 is tending toward 1/√2. 

Answer 1

If for k<n, 2(k+1/2)^2 = (1/2)^2 + (n+1/2)^2 then you will get that cancellation...