Provides a closed-form piecewise quadratic expression for the Frobenius number of shifted squares, obtained via combinatorial reduction, Lagrange's theorem, and generating functions.
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Explicit formulas for F(A) and g(A) are obtained for the semigroup generated by A = (a, ba + d, b²a + ((b²-1)/(b-1))d, ..., b^k a + ((b^k-1)/(b-1))d) when a ≥ k-1 - (d-1)/(b-1), with simplifications for Mersenne, Thabit, repunit, and partial results for Proth semigroups.
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On Frobenius Numbers of Shifted Power Sequences
Provides a closed-form piecewise quadratic expression for the Frobenius number of shifted squares, obtained via combinatorial reduction, Lagrange's theorem, and generating functions.
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On the Frobenius Number and Genus of a Collection of Semigroups Generalizing Repunit Numerical Semigroups
Explicit formulas for F(A) and g(A) are obtained for the semigroup generated by A = (a, ba + d, b²a + ((b²-1)/(b-1))d, ..., b^k a + ((b^k-1)/(b-1))d) when a ≥ k-1 - (d-1)/(b-1), with simplifications for Mersenne, Thabit, repunit, and partial results for Proth semigroups.