Squaring the terms of an ℓth order linear recurrence

is (an(c0, . . . , cl−1; p1, . . . , pl))n≥0, or briefly (an)n≥0, where the pi are constant coefficients, with given aj = cj for all j = 0, 1, . . . , l − 1, and n ≥ l; in such a context, (an)n≥0 is called an l-sequence. In the case l = 2, this sequence is called Horadam’s sequence and was introduced, in 1965, by Horadam [4, 5], and it generalizes many sequences (see [1, 6]). Examples of such sequences are the Fibonacci numbers (Fn)n≥0, the Lucas numbers (Ln)n≥0, and the Pell numbers (Pn)n≥0, when one has the following initial conditions: p1 = p1 = c1 = 1, c0 = 0; p1 = p2 = c1 = 1, c0 = 2; and p1 = 2, p2 = c1 = 1, c0 = 0; respectively. In 1962, Riordan [8] found the generating function for powers of Fibonacci numbers. He proved that the generating function Fk(x) = ∑ n≥0 F k nx n satisfies the recurrence relation