Keith number

In recreational mathematics, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a number in the following integer sequence:

Keith numbers were introduced by Mike Keith in 1987. They are computationally very challenging to find, with only about 100 known.

To determine whether an n-digit number N is a Keith number, create a Fibonacci-like sequence that starts with the n decimal digits of N, putting the most significant digit first. Then continue the sequence, where each subsequent term is the sum of the previous n terms. By definition, N is a Keith number if N appears in the sequence thus constructed.

As an example, consider the 3-digit number N = 197. The sequence goes like this:

Because 197 appears in the sequence, 197 is seen to be indeed a Keith number.

A Keith number is a positive integer N that appears as a term in a linear recurrence relation with initial terms based on its own decimal digits. Given an n-digit number

a sequence $S_{N}$ $S_N$ is formed with initial terms $d_{n-1},d_{n-2},\ldots ,d_{1},d_{0}$ $d_{n-1}, d_{n-2},\ldots, d_1, d_0$ and with a general term produced as the sum of the previous n terms. If the number N appears in the sequence $S_{N}$ $S_N$ , then N is said to be a Keith number. One-digit numbers possess the Keith property trivially, and are usually excluded.

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