Keith Number/Examples/14

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Examples of Keith Number

$14$ is a Keith number:

$1, 4, 5, 9, 14, \ldots$

This sequence is A000285 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 $\left({1, 4}\right)$:

\(\ds K_0\) \(=\) \(\ds 1\)
\(\ds K_1\) \(=\) \(\ds 4\)
\(\ds K_2\) \(=\) \(\ds K_0 + K_1\)
\(\ds \) \(=\) \(\ds 1 + 4\)
\(\ds \) \(=\) \(\ds 5\)
\(\ds K_3\) \(=\) \(\ds K_1 + K_2\)
\(\ds \) \(=\) \(\ds 4 + 5\)
\(\ds \) \(=\) \(\ds 9\)
\(\ds K_4\) \(=\) \(\ds K_2 + K_3\)
\(\ds \) \(=\) \(\ds 5 + 9\)
\(\ds \) \(=\) \(\ds 14\)

Thus $14$ occurs in $K$ and the result follows by definition of Keith number.

$\blacksquare$


Sources