Algorithmic Manipulation of Third-Order Linear Recurrences
Stanley N. Rabinowitz · The Fibonacci Quarterly · 1996
These initial conditions were chosen so that the three sequences form a basis for the set of all third-order linear recurrences with constant coefficients, and because they will allow us (in a future paper) to generalize our results to higher-order recurrences. These three sequences also have nice Binet forms. Given any sequence 〈Sn〉 that satisfies recurrence (1), we can write its elements as a linear combination of Xn, Yn, and Zn, namely Sn = S2Xn + S1Yn + S0Zn. (3)