In this note, we study Fibonacci-like sequences that are defined by the
recurrence <IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img1.gif"
 ALT="$ S_k = a$">, <!-- MATH
 $S_{k + 1} = b$
 -->
<IMG
 WIDTH="75" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.gif"
 ALT="$ S_{k + 1} = b$">, <!-- MATH
 $S_{n + 2} \equiv S_{n + 1} +
S_n$
 -->
<IMG
 WIDTH="149" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.gif"
 ALT="$ S_{n + 2} \equiv S_{n + 1} +
S_n$"> (mod <IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.gif"
 ALT="$ n + 2$">) for all <IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.gif"
 ALT="$ n \geq k$">, where <!-- MATH
 $k, a, b \in \mathbb{N}$
 -->
<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img6.gif"
 ALT="$ k, a, b \in \mathbb{N}$">,
<!-- MATH
 $0 \leq a < k$
 -->
<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img7.gif"
 ALT="$ 0 \leq a &lt; k$">, <!-- MATH
 $0 \leq b < k+ 1$
 -->
<IMG
 WIDTH="115" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.gif"
 ALT="$ 0 \leq b &lt; k+ 1$">, and <!-- MATH
 $(a,b)\ne (0,0)$
 -->
<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img9.gif"
 ALT="$ (a,b)\ne (0,0)$">. We will show
that the number <!-- MATH
 $\alpha = 0.S_k S_{k+1} S_{k+2} \cdots$
 -->
<IMG
 WIDTH="179" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img10.gif"
 ALT="$ \alpha = 0.S_k S_{k+1} S_{k+2} \cdots$"> is irrational.
We also propose a conjecture on the pattern of the sequence <!-- MATH
 $\{S_n\}_{n
\geq k}$
 -->
<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.gif"
 ALT="$ \{S_n\}_{n
\geq k}$">.