论文标题
与解决方案数量相关的移动几何序列的Frobenius编号
The Frobenius number for shifted geometric sequences associated with the number of solutions
论文作者
论文摘要
对于非负整数$ P $，普遍的Frobenius数字之一（称为$ P $ -FROBENIUS数字）是最大的整数，它是最多以$ p $的方式代表的，作为与非阴性整数系数的线性组合，具有一组最大的普通整数隔行器，其最大的普通整数是一组，其最大的普通隔行者是一个。当$ p = 0 $时，Frobenius提出的著名的所谓的Frobenius编号将减少到$ 0 $ -FROBENIUS编号。在19世纪发现了具有两个变量的Frobenius数字的明确公式，但是很难找到具有两个以上变量的公式，并且仅在特殊情况下（例如几何，Thabit，Mersenne等）发现了Frobenius数字的封闭公式。 $ p> 0 $的情况更加困难，并且不知道一个公式。但是，最近，我们终于成功地将$ p $ frobenius的数字作为三角形数字，repunits，fibonacci，fibonacci Triplet和Jacobsthal Triplet的封闭式表达式。 在本文中，我们给出了有限序列$ \ {a b^n-c \} _ n $的$ p $ -frobenius编号的封闭式表达式，其中$ a $，$ b $和$ c $是具有$ a \ ge 1 $，$ b \ ge 2 $ and $ b \ ge 2 $和$ c \ ne 0 $的整数。该序列包括几何，Thabit和Mersenne及其变化的案例。
For a non-negative integer $p$, one of the generalized Frobenius numbers, that is called the $p$-Frobenius number, is the largest integer that is represented at most in $p$ ways as a linear combination with nonnegative integer coefficients of a given set of positive integers whose greatest common divisor is one. The famous so-called Frobenius number proposed by Frobenius is reduced to the $0$-Frobenius number when $p=0$. The explicit formula for the Frobenius number with two variables was found in the 19th century, but a formula with more than two variables is very difficult to find, and closed formulas of Frobenius numbers have been found only in special cases such as geometric, Thabit, Mersenne, and so on. The case of $p>0$ was even more difficult, and not a single formula was known. However, most recently, we have finally succeeded in giving the $p$-Frobenius numbers as closed-form expressions of the triangular number triplet , repunits, Fibonacci triplet and Jacobsthal triplet. In this paper, we give closed-form expressions of the $p$-Frobenius number for the finite sequence $\{a b^n-c\}_n$, where $a$, $b$ and $c$ are integers with $a\ge 1$, $b\ge 2$ and $c\ne 0$. This sequence includes the cases for geometric, Thabit and Mersenne as well as their variations.
