Keith Number/Examples/197

Examples of Keith Number

$197$ is a Keith number:

$1, 9, 7, 17, 33, 57, 107, 197, \ldots$

This sequence is A186830 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 $\tuple {1, 9, 7}$:

\(\ds K_0\)

\(=\)

\(\ds 1\)

\(\ds K_1\)

\(=\)

\(\ds 9\)

\(\ds K_2\)

\(=\)

\(\ds 7\)

\(\ds K_3\)

\(=\)

\(\ds K_0 + K_1 + K_2\)

\(\ds \)

\(=\)

\(\ds 1 + 9 + 7\)

\(\ds \)

\(=\)

\(\ds 17\)

\(\ds K_4\)

\(=\)

\(\ds K_1 + K_2 + K_3\)

\(\ds \)

\(=\)

\(\ds 9 + 7 + 17\)

\(\ds \)

\(=\)

\(\ds 33\)

\(\ds K_5\)

\(=\)

\(\ds K_2 + K_3 + K_4\)

\(\ds \)

\(=\)

\(\ds 7 + 17 + 33\)

\(\ds \)

\(=\)

\(\ds 57\)

\(\ds K_6\)

\(=\)

\(\ds K_3 + K_4 + K_5\)

\(\ds \)

\(=\)

\(\ds 17 + 33 + 57\)

\(\ds \)

\(=\)

\(\ds 107\)

\(\ds K_7\)

\(=\)

\(\ds K_4 + K_5 + K_6\)

\(\ds \)

\(=\)

\(\ds 33 + 57 + 107\)

\(\ds \)

\(=\)

\(\ds 197\)

Thus $197$ occurs in $K$ and the result follows by definition of Keith number.

$\blacksquare$