The Jacobsthal-Padovan-Fibonacci <i>p</i>-sequence and its application in the concise representation of vibrating systems with dual proximal groups

The focus of this paper is on the Jacobsthal-Padovan-Fibonacci (JPF) p-sequence {JPa(n)(F,p)}, p >= 3, n>0 and the connections between the Fibonacci and Jacobsthal-Padovan p-sequences. Also, a new Binet formula and a new combinatorial representation as well as some useful properties of the JPF p-numbers are given. The main results are an exponential form of JPa(n)(F,p), useful identities for m >= p+4, P-m,p(JPa,F), R(m,p)(JPa,F )and for m>p+4, S-m,p(JPa,F), topologies for every descriptive proximity space, with application in dual proximal group representations of the topology on a proximity space on the Hilbert envelope lobes on a vibrating system waveform