998,001 = 999² and the decimal expansion of 1/n² for n = 10^k - 1 generates all k-digit strings in sequence. The same trick works for 1/9801 = 1/99² which cycles through all two-digit sequences 00-98 (skipping 99).
The mechanism: the numerator of 1/(n-1)² expands via the geometric series, and the carrying behavior at the boundary creates the one missing entry.
A fun extension: 1/99² × 99 = 1/99 = 0.010101... where the pattern "collapses" because you're now looking at the simpler 1/(10² - 1) structure. The richer pattern only emerges for the squared denominator.
998,001 = 999² and the decimal expansion of 1/n² for n = 10^k - 1 generates all k-digit strings in sequence. The same trick works for 1/9801 = 1/99² which cycles through all two-digit sequences 00-98 (skipping 99).
The mechanism: the numerator of 1/(n-1)² expands via the geometric series, and the carrying behavior at the boundary creates the one missing entry.
A fun extension: 1/99² × 99 = 1/99 = 0.010101... where the pattern "collapses" because you're now looking at the simpler 1/(10² - 1) structure. The richer pattern only emerges for the squared denominator.