One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer recurrence sequences as rational polynomial linear combinations of Fibonacci numbers.

The discovery of high-density liquid (HDL) and low-density liquid (LDL) water has been a major success of molecular simulations, yet extending this analysis to interfacial water is challenging due to conventional order parameters assuming local homogeneity. This limitation previously prevented resolving the composition of the surface layer of the liquid/vapour interface. Here, we apply a recently introduced topological order parameter [R. Foffi and F. Sciortino, J. Phys. Chem. B 127, 378-386 (2022)] to analyze the composition of the water/vapor interface across a broad temperature range. Our results reveal that LDL-like water dominates the outermost region at all temperatures, while HDL-like water accumulates beneath it, presenting a clear layering roughly below the temperature of maximum density. This structured stratification, previously inaccessible, highlights the power of the topological order parameter in resolving interfacial molecular heterogeneity and provides new insights into the structural properties of water at interfaces.

The Hilbert basis is fundamental in describing the structure of the integer points of a polyhedral cone. The face-centered cubic grid is one of the densest packing of the 3-dimensional space. The cycles of a grid satisfy the constraint set of a pointed, polyhedral cone which contains only non-negative integer vectors. The Hilbert basis of a grid gives the structure of the basic cycles in the grid. It is shown in this paper that the basic cycles of the FCC grid belong to 11 types. It is also discussed that how many elements are contained in the individual types. The proofs of the paper use geometric, combinatorial, algebraic, and operations research methods.