Balancing and Lucas-Balancing Numbers and their Application to Cryptography

It is well known that, a recursive relation for the sequence  is an equation that relates  to certain of its preceding terms . Initial conditions for the sequence  are explicitly given values for a finite number of the terms of the sequence. The recurrence relation is useful in certain counting problems like Fibonacci numbers, Lucas numbers, balancing numbers, Lucas-balancing numbers etc. In this study, we use the recurrence relations for both balancing and Lucas-balancing numbers and examine their application to cryptography.