The Infinite Square Root (and Golden Ratio at q == 1)

#!/bin/env python3

# Infinite square root: x = √(q + √(q + √(q + …)))
# Basic equation: 0 = x² − x − q
# Higher order equation: 0 = xᵃ − bx − c

Q_MAX = 6  # highest q to calculate
EXPONENT_MAX = 6  # highest log₂(a) to calculate
LINEAR_MAX = 2 ** EXPONENT_MAX

def exponential_next(q):
  return lambda a, b, c: (2 * a, b**2 + 2 * b * c, q * b**2 + c**2)

def linear_next(q):
  return lambda a, b, c: (2 + a, b * (q + 1) + c, q * (b + c))

def single_next(q):
  return lambda a, b, c: (1 + a, b + c, q * b)

def typeset(a, b, c):
  n = ''.join('⁰¹²³⁴⁵⁶⁷⁸⁹'[ord(d) - ord('0')] for d in str(a))
  return f'0 = x{n} − {b if b != 1 else ""}x − {c}'

def print_series(q, nsteps, step):
  a = (2, 1, q)
  next_a = step(q)
  while a[0] < nsteps:
    print(typeset(*a))
    a = next_a(*a)
  print(typeset(*a))

def print_quotient(q):
  print(f'x = √({q} + √({q} + √({q} + …)))')
  print(f'x² = {q} + √({q} + √({q} + …))')
  print()
  for note, step in (('Exponential:', exponential_next),
                     ('Linear by 2:', linear_next),
                     ('Linear by single step:', single_next)):
    print(note)
    print_series(q, LINEAR_MAX, step)

for q in range(1, Q_MAX):
  print_quotient(q)
  print()
print_quotient(Q_MAX)