We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4, 6+4k, 87−4k) and S(9, 3+9k, 85−9k) generated by three elements. We present
generalization of these sequences by numerical semigroups S(r2 1 ,r1r2 + r2 1 k,r3 − r2 1 k), k ∈ Z, r1,r2,r3 ∈ Z+, r1 ≥ 2 and gcd(r1,r2) = gcd(r1,r3) = 1, and calculate their universal Frobenius number Φ(r1,r2,r3) for the wide range of k providing semigroups be symmetric. We show that this type of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r2 1 ,r3 − r2 1 k} for sporadic values of k and find these values by solving the quadratic Diophantine equation.

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