Generalized Schur Numbers for x1 + x2 + c=3x3

Let r(c) be the least positive integer n such that every two coloring of the integers 1,..., n contains a monochromatic solution to x(1)+x(2)+c = 3x(3).Verifying a conjecture of Martinelli and Schaal, we prove that r(c) = inverted right perpendicular 2inverted right perpendicular2 + c/3inverterted left perpendicular + c/3inverterted left perpendicular for all c >= 13, and r(c) = inverted right perpendicular 3inverted right perpendicular3-c/2inverterted left perpendicular - c/2 inverterted left perpendicular, for all c <= -4.