Let {ai, j} be real numbers for 0 ≤ i ≤ r - 1 and 1 ≤ j ≤ 2, and define a sequence {vn} with initial conditions v0, v1 and conditional linear recurrence relation vn = at,1Vn-1 + at,2Vn-2 where n ≡ t (mod r) (n ≥ 2). The sequence {vn} can be viewed as a generalization of many well-known integer sequences, such as Fibonacci, Lucas, Pell, Jacobsthal, etc. We find explicitly a linear recurrence equation which is satisfied by {vn}, generating functions, matrix representations and extended Binet’s formulas for {vn} in terms of a generalized continuant.

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Persistent URL dx.doi.org/10.1515/9783110298161.1042
Panario, D, Sahin, M. (Murat), & Wang, Q. (2014). A family of fibonacci-like conditional sequences. In Integers: Annual Volume 2013 (pp. 1042–1055). doi:10.1515/9783110298161.1042