Keith number

In number theory, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a natural number formula_1 in a given number base formula_2 with formula_3 digits such that when a sequence is created such that the first formula_3 terms are the formula_3 digits of formula_1 and each subsequent term is the sum of the previous formula_3 terms, formula_1 is part of the sequence. Keith numbers were introduced by Mike Keith in 1987.
They are computationally very challenging to find, with only about 100 known. 
Definition.
Let formula_1 be a natural number, let formula_10 be the number of digits in the number in base formula_2, and let 
be the value of each digit of the number.
We define a linear recurrence relation formula_13 such that for formula_14, 
and for formula_16
If there exists an formula_18 such that formula_19, then formula_1 is said to be a Keith number. 
For example, 88 is a Keith number in base 6, as 
and the entire sequence
and formula_25.
Finding Keith numbers.
Whether or not there are infinitely many Keith numbers in a particular base formula_2 is currently a matter of speculation. Keith numbers are rare and hard to find. They can be found by exhaustive search, and no more efficient algorithm is known.
According to Keith, in base 10, on average formula_27 Keith numbers are expected between successive powers of 10. Known results seem to support this.
Examples.
14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 62662, 86935, 93993, 120284, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993, 1084051, 7913837, 11436171, 33445755, 44121607, 129572008, 251133297, ... 
Other bases.
In base 2, there exists a method to construct all Keith numbers. 
The Keith numbers in base 12, written in base 12, are
Keith clusters.
A Keith cluster is a related set of Keith numbers such that one is a multiple of another. For example, in base 10, formula_28, formula_29, and formula_30 are all Keith clusters. These are possibly the only three examples of a Keith cluster in base 10.
Programming example.
The example below implements the sequence defined above in Python to determine if a number in a particular base is a Keith number:
def is_repfigit(x: int, b: int) -> bool:
 """Determine if a number in a particular base is a Keith number."""
 if x == 0:
 return True
 sequence = []
 y = x
 while y > 0:
 sequence.append(y % b)
 y = y // b
 digit_count = len(sequence)
 sequence.reverse()
 while sequence[len(sequence) - 1] < x:
 n = 0
 for i in range(0, digit_count):
 n = n + sequence[len(sequence) - digit_count + i]
 sequence.append(n)
 return (sequence[len(sequence) - 1] == x)