Definitions

• The $k$-Fibonacci-$q$ sequence (the $n$th term of which is denoted $F^k_{n,q}$) is defined recursively by the following statements:
  • If $2-k\leq n\leq 0, n\in{\bf Z}$, then $F_{n,q}^k=0$.
  • $F_{1,q}^k=q$.
  • $F_{n,q}^k=\sum_{i=1}^k F_{n-i,q}^k$, $n>0$.
  Note that this idea encompasses both the $k$-Fibonacci sequence and the Fibonacci-$q$ sequence.
• $z^k_q (a)=c$ means that $c$ has the following properties:
  • $c>0$.
  • For all $i$ such that $c-(k-2)\leq i\leq c, i\in{\bf Z}$, it follows that $F_{i,q}^k\equiv 0\pmod{a}$.
  • $c$ is the least positive integer satisfying these properties.
• $t^k_q (a)=d$ means that $d$ has the following properties:
  • $d>0$
  • For all $i$ such that $d-(k-2)\leq i\leq d, i\in{\bf Z}$, it follows that $F_{i,q}^k\equiv 0\pmod{a}$.
  • $F_{d+1,q}^k\equiv q\pmod{a}$.
  • $d$ is the least positive integer satisfying these properties.
  Note: $t^k_q (a)$ is then the function that gives the period of the $k$-Fibonacci-$q$ sequence in modulo $a$.
• $numz^k (a) = g$ means that $g$ has the following properties:
  • $g$ is the number of $n$ that satisfy the property that all $i$ from $n-(k-2)$ to $n$, inclusive, satisfy the property that $F_i^k\equiv 0\pmod{a}$, where $n$ goes from $1$ to $t^k (a)$, inclusive.
• $numz^k_q (a) = h$ means that $h$ has the following properties:
  • $h$ is the number of $n$ that satisfy the property that all $i$ from $n-(k-2)$ to $n$, inclusive, satisfy the property that $F_{i,q}^k\equiv 0\pmod{a}$, where $n$ goes from $1$ to $t^k_q (a)$, inclusive.