  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.
