Lucas resonances of the three planet system TOI-1749

The three exoplanets of the TOI-1749 system are labelled b, c and d. A year ago, an article in the Astronomical Journal on this system by a group of scientists noted a near 2:1 orbital ratio of planets c and d, but made no reference to synodic periods.

Now, with newer data from exoplanet.eu, we analyse the synodics. The mean period between alignments of two planets with their star is as follows:
b-c = 5.0989527 days
c-d = 8.922796
b-d = 3.2447389

From that, the ratios can be obtained:
7 b-c = 35.692668 days
4 c-d = 35.691184
11 b-d = 35.692127

4:7:11 coincides with the Lucas number series which is closely related to the Fibonacci series and the golden ratio.

Since the numbers of the planetary orbits are not whole numbers in that period (although close to 15b, 8c and 4d), a factor of 18 has been used to get a more accurate picture (see chart).

269 b = 642.477 days
143 c = 642.485
71 d = 642.529

126 b-c = 642.468 days
72 c-d = 642.441
198 b-d = 642.458

A simple comparison with the 4-7-11 calculations (above):
35.692 * 18 = 642.456