Fibodiv Numbers

My diurnal age today is 27321 and I was struggling to find an interesting sequence to which this number belonged. Fortunately, Numbers Aplenty came to my aid with the information that 27321 is a fibodiv number. Such numbers are few and far between. Here's the definition that is provided by the described source:

These are numbers \(n\)  whose representation can be split into two numbers, say \(a\) and  \(b\), such that the Fibonacci-like sequence which uses \(a\) and \(b\) as seeds contains \(n\)  itself.

 In the case of 27321, it can be seen that this is indeed the case:

273, 21, 294, 315, 609, 924, 1533, 2457, 3990, 6447, 10437, 16884, 27321

The sequence of such numbers can be found in the OEIS A130792 but they are not referred to as fibodiv numbers. 

A130792

Numbers \(n\) whose representation can be split in two parts which can be used as seeds for a Fibonacci-like sequence containing \(n\) itself.

The initial members are:

14, 19, 28, 47, 61, 75, 122, 149, 183, 199, 244, 298, 305, 323, 366, 427, 488, 497, 549, 646, 795, 911, 969, 1292, 1301, 1499, 1822, 1999, 2087, 2602, 2733, 2998, 3089, 3248, 3379, 3644, 3903, 4555, 4997, 5204, 5466, 6178, 6377, 6496, 6505, 7288, 7806, 7995 

Between 20,000 and 40,000, the numbers are:

19999, 20816, 20987, 21623, 22117, 23418, 24712, 24719, 26020, 27321, 27483, 27801, 28622, 29107, 29923, 29998, 30890, 31224, 32498, 32525, 33826, 33979, 35127, 36428, 36644, 37729, 39030

As can be seen, there's quite a gap between 27321 and the previous sequence member, 26020, but the subsequent member, 27483, is much closer. All the numbers in the sequence admit of only one concatenation and it is not known if there are numbers that admit of more than one.

The sequence is infinite since 19 with seeds 1 and 9, 199 with seeds 1 and 99, 1999 with seeds 1 and 999, 19999 with seeds 1 and 9999 and so on are in the sequence. For example:

1, 9999, 10000, 19999

It's easy enough to confirm that a number is a fibodiv (see permalink) but finding them initially is more challenging. However, OEIS comments list a method but I don't quite understand it.