The three basis sequences A, B and C

We study the recurrence f(1) = γ, f(2n) = 2f(n) + β₀ and f(2n + 1) = 2f(n) + β₁ for n ≥ 1, where γ, β₀ and β₁ are integers. Taking inspiration from part III, make explicit the three sequences A(n), B(n) and C(n) associated respectively with the coefficient triples (1, 0, 0), (0, 1, 0) and (0, 0, 1). Express them in terms of n and of α, the largest exponent in the base-2 expansion of n. (Question IV.1.a of the assignment.)