According to this answer transits of Earth across the disk of the Sun as seen from Mars:
The (Arthur C.) Clarke story is quite correct. Earth (and Moon) would transit the sun from the viewpoint of Mars on 8 May 1905, 11 May 1984 and again on 10 Nov 2084.
no surprises there, but they
...follow a pattern of occurring after 26, 79 then 100 years.
Why is that exactly? Why this pattern?
Such transit events occur when Earth and Mars are simultaneously at a node where their orbital planes intersect.
Between transits at the same node, both Earth and Mars must complete whole numbers of orbits, where 1 Mars year = 1.8808 Earth years. Meeus and Goffin 1983 identify patterns of 79-year and 284-year intervals:
79.0 Earth years ≈ 42.0 Mars years
284.0 Earth years ≈ 151.0 Mars years
Their Table II arranges the May and November transit dates in separate "panoramas" with 284 years between rows and 79 years between columns. These are somewhat analogous to a saros-inex panorama of solar eclipses.
Between transits at opposite nodes, both Earth and Mars must complete odd numbers of half-orbits:
Nov→May: 25.5 Earth years ≈ 13.56 Mars years
May→Nov: 100.5 Earth years ≈ 53.43 Mars years
These Mars year fractions differ from 0.5 because the half-orbit near perihelion takes significantly less time than the half near aphelion. Meeus and Goffin Table I also shows some intervals of 21.5 and 53.5 years outside the present few centuries.
