The Reuleaux Triangle Revisited

In all my posts since the middle of 2015, I've only mentioned the Reuleaux (pronounciation) Triangle once and there's a good reason for this. Let's look at my original post from the 9th of November 2021 titled Reuleaux Triangle. I wrote (text in blue):

Today, having turned 26518 days old, I found an interesting property of this number that qualifies it for membership in OEIS A340644:

A340644

The number of vertices on a Reuleaux triangle formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

 The initial members of the sequence are:

3, 19, 120, 442, 1332, 2863, 5871, 10171, 17358, 26518, 40590, 57757, 81735, 110209, 148158, 192184, 248772, 313105, 393429, 483283, 593490, 715528, 861660, 1022281, 1211811, 1418515, 1659108, 1919842, 2220204, 2543527, 2912751, 3308305, 3755922, 4233730, 4770150, 5340529, 5977071 

26518 corresponds to the case where \(n\) = 10 and this number stands in splendid isolation from its neighbours: 17358 (\(n\) = 9) and 40590 (\(n \) = 11). The former occurred before I started monitoring my diurnal age and the second will occur long after I'm dead. In my post, I included the image shown in Figure 1.

Figure 1

The post I made back then is a good one, if I do say so myself, and delves into some interesting properties and applications of the Reuleaux triangle. The reason the triangle has popped up again has to do again with my diurnal age (27336) but this time it's regions rather than vertices that are involved. The number is a member of OEIS A340639:

A340639

The number of regions inside a Reuleaux triangle formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

The initial members of the sequence are:

1, 24, 145, 516, 1432, 3084, 6106, 10638, 17764, 27336, 41233, 58902, 82675, 111864, 149497, 194430, 250534, 316020, 395728, 487122, 596434, 720162, 865321, 1027974, 1216291, 1425348, 1664539, 1928022, 2226658, 2553204, 2920378, 3319536, 3764848, 4246638, 4780489, 5355414, 5988973 

As with the number mentioned in the earlier post, 27336 corresponds to the case of \(n\) = 10 and once again its neighbours (17764 and 41233) are far removed. The OEIS comments provide a link to a very colourful image of the regions. See Figure 2.

Figure 2: source

However, the regions can be seen more clearly perhaps in the case of \(n\) = 2. See Figure 3.

Figure 3: source

So it may be some time before the Reuleaux triangle gets a mention again because the only integers associated with it seem to relate to vertices and regions arising from the division of its sides into \(n\) equal parts.