Calculating 2-adic square roots

A sequence of positive integers a_n < 2ⁿ⁺², n ≥ 0 is constructed, which converges 2-adically to a 2-adic square root of a:
a_n-1² ≡ a (mod 2ⁿ⁺²),
a₀ = 1 or 3, a_n = a_n-1 + b_n2ⁿ⁺¹, n ≥ 1.
b_n is 0 or 1 and is determined by b_n ≡ (a_n-1² - a)/2ⁿ⁺² (mod 2).
a₀ = 1 implies x = 1 +0*2 + b₁2² + ···
a₀ = 3 implies x = 3 +1*2 + b₁2² + ···

(See lecture notes and solutions).