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Balancing and Lucas-balancing numbers are solutions of a Diophantine equationand satisfy a second order homogeneous recurrence relation. Interestingly,these numbers can be seen as numerators and denominators in the steady stateprobabilities of a class of transition probability matrices of Markov chains.An identity relating the balancing numbers and the silver ratio can be obtainedas a byproduct.

Appearance of Balancing and related number sequences in steady state probabilities of some Markov chains