Construction of Goethals-Seidel sequences, 3

Divide and concur searches for Goethals-Seidel sequences are described. Focus is on the  "twin case"
form 2x^2+y^2+z^2 that represents numbers 4t, for t odd. [The smallest open case is t=167.] We both describe and carefully
listen to suggestions of future steps that might bring progress toward the solution of this problem.