Combinatorial numbers in binary recurrences

Diophantine Equations Related with Linear Binary Recurrences

In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...

Combinatorial Recurrences and Linear Difference Equations

In this work we introduce the triangular arrays of depth greater than 1 given by linear recurrences, that generalize some well-known recurrences that appear in enumerative combinatorics. In particular, we focussed on triangular arrays of depth 2, since they are closely related to the solution of linear three–term recurrences. We show through some simple examples how these triangular arrays appe...

A combinatorial solution to intertwined recurrences

Second-order Linear Recurrences of Composite Numbers

In a well-known result, Ronald Graham found a Fibonacci-like sequence whose two initial terms are relatively prime and which consists only of composite integers. We generalize this result to nondegenerate second-order recurrences.

Binary Combinatorial Coding

We present a novel binary entropy code called combinatorial coding (CC). The theoretical basis for CC has been described previously under the context of universal coding [1], enumerative coding [2], and minimum description length [3]. The code described in these references works as follows: assume the source data of length M is binary, memoryless, and generated with an unknown parameter θ, the ...