Can the sum of 2 consecutive positive odd integers each squared be a Triangular number?

Analyzing the equation (2n-1)^2+(2n+1)^2=k(k+1)/2 for natural number solutions was how this slightly modified Problem #224 from Sierpinski's 250 problem Number theory tome. Math expert and Harvard graduate Gerry Myerson answered problem from this text September 2014. Roel Van de Paar coverage of free variables is instructive. January 2021. Thanks to Sean Morrell @s34nm94 for his interest and contributions. The factoring mistake he made is quite common even among experienced tutors, teachers and chess players with a predilection for math.
Sierpinski problem mistakenly stated 2 consecutive positive integers. Could not find errata anywhere.
University of Pittsburgh Mathematician John Parker work and analysis a must read!