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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $14$