It is known that the size sequence of diatoms in a chain-like colony can be described by a Lindenmayer system (L-system), provided that the vegetative divisions are synchronous. From this L-system, one can derive a simple calculation scheme for the size sequences. (Details can be found here: https://diatoms.de/en/chain-like-colonies)
To check whether a found fragment of a chain satisfies these regularities, it is helpful to consider the sequence of the differences of the size indices. That sequence can be interpreted geometrically as a dragon curve. This reveals a surprising connection between the biology of chain-like diatom colonies and the mathematics of fractals (see: https://diatoms.de/en/size-sequence-in-chains-ii).
In much greater depth and with formal mathematical proofs, the issues can be found in the following paper:
Harbich, T. (2021), On the Size Sequence of Diatoms in Clonal Chains in Diatom Morphogenesis (Diatoms: Biology and Applications) Vadim V. Annenkov (Editor), Richard Gordon (Editor), Joseph Seckbach (Editor), Wiley-Scrivener; 1. Edition
