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Let (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f, g : S → K of the functional equation f(x+y)+f(x+σ(y))+g(x+y)=2f(x)+g(x)g(y)     for  x,y∈S. $$f(x+y)+f(x+\sigma (y\left)\right)+g(x+y)=2f\left(x\right)+g\left(x\right)g\left(y\right)\mathrm{for}x,y\in S.$$ We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy...