论文标题
$ p $ -frobenius和$ p $ -sylvester的斐波那契和卢卡斯三胞胎的数字
The $p$-Frobenius and $p$-Sylvester numbers for Fibonacci and Lucas triplets
论文作者
论文摘要
在本文中，我们研究了某种弗罗贝尼乌斯的某种广义线性二磷剂问题。令$ a_1，a_2，\ dots，a_l $为正整数，以使它们最大的常见除数是一个。 For a nonnegative integer $p$, denote the $p$-Frobenius number by $g_p(a_1,a_2,\dots,a_l)$, which is the largest integer that can be represented at most $p$ ways by a linear combination with nonnegative integer coefficients of $a_1,a_2,\dots,a_l$.当$ p = 0 $，$ 0 $ -FROBENIUS编号是经典的Frobenius号码。当$ l = 2 $，$ p $ -frobenius号码明确给出。但是，当$ l = 3 $甚至更大时，即使在特殊情况下，明确给Frobenius编号也不容易，而且当$ p> 0 $时，它甚至更加困难，而且尚无具体的示例。但是，最近，我们成功地为$ l = 3 $的情况下的序列是三角形数量或对偿还的情况提供了明确的公式。在本文中，我们在$ p> 0 $时显示了fibonacci三倍的明确公式。此外，我们给出了$ p $ -Sylvester号码的明确公式，即，最多可以以$ p $方式表示的非负整数总数。此外，显示了有关卢卡斯三重的明确公式。
In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $a_1,a_2,\dots,a_l$ be positive integers such that their greatest common divisor is one. For a nonnegative integer $p$, denote the $p$-Frobenius number by $g_p(a_1,a_2,\dots,a_l)$, which is the largest integer that can be represented at most $p$ ways by a linear combination with nonnegative integer coefficients of $a_1,a_2,\dots,a_l$. When $p=0$, $0$-Frobenius number is the classical Frobenius number. When $l=2$, $p$-Frobenius number is explicitly given. However, when $l=3$ and even larger, even in special cases, it is not easy to give the Frobenius number explicitly, and it is even more difficult when $p>0$, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers or of repunits for the case where $l=3$. In this paper, we show the explicit formula for the Fibonacci triple when $p>0$. In addition, we give an explicit formula for the $p$-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most $p$ ways. Furthermore, explicit formulas are shown concerning the Lucas triple.
