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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5 Pith papers cite this work. Polarity classification is still indexing.
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UNVERDICTED 5representative citing papers
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.
Reduces Apéry set of semigroup quotients to minimization when p divides a1, giving closed Frobenius formulas for almost arithmetic progression generators and partially solving a prior open problem.
A combinatorial reduction of the Frobenius problem to an optimization task produces explicit formulas for g(A), n(A), and s(A) on special sequences and applies MacMahon's partition analysis to count representations.
Extends the stable property of Frobenius numbers to sequences A(a)=(a, ha+dB) yielding a congruence-class characterization of g(A(a)) mod bk for large a, plus explicit formulas for several B.
citing papers explorer
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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.
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On quotients of numerical semigroups for almost arithmetic progressions
Reduces Apéry set of semigroup quotients to minimization when p divides a1, giving closed Frobenius formulas for almost arithmetic progression generators and partially solving a prior open problem.
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A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)
A combinatorial reduction of the Frobenius problem to an optimization task produces explicit formulas for g(A), n(A), and s(A) on special sequences and applies MacMahon's partition analysis to count representations.
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The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$
Extends the stable property of Frobenius numbers to sequences A(a)=(a, ha+dB) yielding a congruence-class characterization of g(A(a)) mod bk for large a, plus explicit formulas for several B.