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.
On Frobenius Numbers of Shifted Power Sequences
3 Pith papers cite this work. Polarity classification is still indexing.
3
Pith papers citing it
abstract
We resolve the open problem of characterizing the Frobenius number $g(A)$ for shifted square sequences $A = (a, a+1^2, \ldots, a+k^2)$, confirming a conjecture of Einstein et al. (2007). By combining a combinatorial reduction to an optimization problem with Lagrange's Four-Square Theorem and generating function techniques, we derive an explicit formula for $g(A)$: a piecewise quadratic polynomial in $a$, classified by residue classes modulo $k^2$.
years
2023 3verdicts
UNVERDICTED 3representative citing papers
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
-
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.
-
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.
-
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.