Formulas for Fibonacci-Like Sequences Produced by Pascal-Like Triangles

In this paper we are going to present three formulas to express Fibonacci-like sequences with the Fibonacci sequence. We constructed Pascal-like triangles using probabilities of a game, and these Pascal-like triangles can be considered generalizations of the well known Pascal\u27s triangle. Using these triangles, we can make Fibonacci-like sequences. We discovered an interesting formula for these Fibonacci-like sequences. We could generalize this formula, and got some interesting formulas that combine these Fibonacci-like sequences and the Fibonacci-sequence. These formulas can reveal very interesting relationships between Fibonacci-like sequences and the Fibonacci sequence, and we can expect a rich possibility of the research from these Fibonacci-like sequences

Rational strategy for shaped nanomaterial synthesis in reverse micelle reactors

The shape-controlled synthesis of nanoparticles was established in single-phase solutions by controlling growth directions of crystalline facets on seed nanocrystals kinetically; however, it was difficult to rationally predict and design nanoparticle shapes. Here we introduce a methodology to fabricate nanoparticles in smaller sizes by evolving shapes thermodynamically. This strategy enables a more rational approach to fabricate shaped nanoparticles by etching specific positions of atoms on facets of seed nanocrystals in reverse micelle reactors where the surface energy gradient induces desorption of atoms on specific locations on the seed surfaces. From seeds of 12 nm palladium nanocubes, the shape is evolved to concave nanocubes and finally hollow nanocages in the size ~10 nm by etching the center of {200} facets. The high surface area-to-volume ratio and the exposure of a large number of palladium atoms on ledge and kink sites of hollow nanocages are advantageous to enhance catalytic activity and recyclability