Redundancy in Fibonacci Numbers: A Thought Experiment (Fibonacci Numbers & Atoms: A Thought Experiment)

An investigation of the properties of numbers generated according to recursive equations in general, including equations giving rise to the Fibonacci Sequence and the Lucas Sequence. Examples are also provided of Excel (RTM) worksheets by which various recursive number sequences may be generated, and redundancy in how different vlaues of a sequnce may be achieved. Properties of the Cassini Identity and Catalan Identity are investigated, and the illustration of how values of all recursive number sequences are linked to the Standard Fibonacci Sequence by a so-called "Cassini Coefficient" of F(0)-squared minus F(1)*F(-1). This coefficient gives a value of "5" for values of the Lucas Sequence and "-1" for values of the Standard Fibonacci Sequence. Also investigated are Complex properties of both the Standard Fibonacci Sequence and the Lucas Sequence, values of each element of the two sequences being reproduced by the sum of squares comprising both a Real component and an Imaginary component. Also investigated are properties of values of numbers generated through repetitive multiplication in modulo counter systems, with multipliers being selected based on the Cassini Identity. Finally it is investigated whether properties of both the Cassini Identity and the Catalan Identity, in assuming an Aether-type environment allowing electro-magnetic propogation, may provide a different model for the electronic structure of atoms.