Simple continued fraction

A (simple) continued fraction for a real number $𝛼 \in ℝ$ is an expression of the form

𝛼 = [𝑎 0; 𝑎 1, 𝑎 2, \dots] = 𝑎 0 + 1 𝑎 1 + 1 𝑎 2 + \dots .

where $𝑎 𝑖 \in ℤ$ , More precisely, $𝛼 = l i m 𝑛 \to \infty 𝛼 𝑛$ , where the $𝑛 t h$ convergent

𝛼 𝑛 = [𝑎 0; 𝑎 1, \dots, 𝑎 𝑛] = 𝑝 𝑛 𝑞 𝑛

where there are defined by the recurrence relations

𝑝 - 2 = 0, 𝑝 - 1 = 1, 𝑝 𝑛 = 𝑎 𝑛 𝑝 𝑛 - 1 + 𝑝 𝑛 - 2 𝑞 - 2 = 1, 𝑞 - 1 = 𝑎 0, 𝑞 𝑛 = 𝑎 𝑛 𝑞 𝑛 - 1 + 𝑞 𝑛 - 2

Sage

The methods .p and .q correspond to the notation given above

K.<α> = QuadraticField(223)
continued_fraction(α) 
# [14; (1, 13, 1, 28)*]
continued_fraction(α).p(1) 
# 209
continued_fraction(α).q(1) 
# 14
continued_fraction(α).convergent(2) 
# 209/14

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