Keith Number/Examples/19

Examples of Keith Number

$19$ is a Keith number:

$1, 9, 10, 19, \ldots$

This sequence is A022099 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\left({1, 9}\right)$:

\(\ds K_0\)

\(=\)

\(\ds 1\)

\(\ds K_1\)

\(=\)

\(\ds 9\)

\(\ds K_2\)

\(=\)

\(\ds K_0 + K_1\)

\(\ds \)

\(=\)

\(\ds 1 + 9\)

\(\ds \)

\(=\)

\(\ds 10\)

\(\ds K_3\)

\(=\)

\(\ds K_1 + K_2\)

\(\ds \)

\(=\)

\(\ds 9 + 10\)

\(\ds \)

\(=\)

\(\ds 19\)

Thus $19$ occurs in $K$ and the result follows by definition of Keith number.

$\blacksquare$