Keith Number/Examples/251,133,297

Examples of Keith Number
$251 \, 133 \, 297$ is a Keith number:
 * $2, 5, 1, 1, 3, 3, 2, 9, 7, 33, 64, 123, 245, 489, 975, 1947, 3892, 7775,$
 * $15 \, 543, 31 \, 053, 62 \, 042, 123 \, 961, 247 \, 677, 494 \, 865, 988 \, 755,$
 * $1 \, 975 \, 563, 3 \, 947 \, 234, 7 \, 886 \, 693, 15 \, 757 \, 843, 31 \, 484 \, 633,$
 * $62 \, 907 \, 224, 125 \, 690 \, 487, 251 \, 133 \, 297, \ldots$

Proof
By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\left({2, 5, 1, 1, 3, 3, 2, 9, 7}\right)$:

Thus $251 \, 233 \, 297$ occurs in $K$ and the result follows by definition of Keith number.