Sequence Formed From Digit Display Elements

In my post titled Polyominoes and Conway's Game of Life (February 19th 2024), I looked at the representation of the digits 0, 1 and 2 as polyominoes. In a subsequent post titled Digits 3 to 9 in Conway's Game of Life (February 20th 2024), I examined the digits from 3 to 9 in the same light. Somewhat earlier, in a post titled Diurnal Age Meets Conway's Game Of Life (February 15th 2024), I began to investigate how the number associated with my diurnal age behaves under the Game of Life rules and since 27346 I've been doing this on a daily basis. The results I've been recording in my Airtable database.

My diurnal age today is 27380 and in terms of polyominoes it looks as shown in Figure 1:

Figure 1

This representation uses 54 squares and it occurred to me that starting from 0 and progressing through the natural numbers, records will be set for the number of squares required to represent the numbers. So I set out to determine these record number of squares and the numbers with which they were associated. 

The first step was to set up a data dictionary linking each digit with the number of squares in its polyomino. The dictionary looks like this with digit first followed by the number of squares:

{0:12, 1:5, 2:11, 3:11, 4:8, 5:11, 6:12, 7:7, 8:13, 9:12}

The results in the range from 0 to 100000 are shown in the table in Figure 2 (permalink).

Figure 2

Putting the results in list format, we have the following records:

12, 13, 17, 18, 23, 24, 25, 26, 29, 30, 31, 35, 36, 37, 38, 39, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65

The numbers associated with these records are:

0, 8, 10, 18, 20, 28, 68, 88, 100, 108, 188, 200, 208, 288, 688, 888, 1000, 1008, 1088, 1888, 2000, 2008, 2088, 2888, 6888, 8888, 10000, 10008, 10088, 10888, 18888, 20000, 20008, 20088, 20888, 28888, 68888, 88888

Surprisingly these numbers make an appearance in OEIS A143617:

A143617

Where record values occur in A010371: number of segments used to represent n on a 7-segment calculator display.

The record values are different since in OEIS A010371 we are counting dashes and not squares. It's the numbers at which these records occur that are the same. The calculator display digits are shown in Figure 3:

Figure 3

Looking at Figure 2 it can be seen that my square total of 54 for today's number of 27380 was reached for the first time way back in 10008. Even though it would be much more labour intensive, another sequence could be developed that counts that number of generations required for a number to reach stability under Conway's Game of Life rules. 

For example, 27380 requires about 380 generations to reach the stable configuration shown in Figures 4 and 5.

Figure 4

Figure 5

The single "toad" and two "traffic lights" alternate between the shapes shown in the two figures whereas the still life "blocks" (two of them), the "pond" (one of them) and the single "honey farm" (the group of four "beehives") remain the same. There's no way of telling how many generations are required for each number to reach stability and so they would all need to be tested individually.