prove_recurrence.py

#!/usr/bin/env python3
"""Symbolic induction step for the exact t=3 vector-sector sequences."""

import sys
from pathlib import Path

if Path("/private/tmp/bb_sympy").is_dir():
    sys.path.insert(0, "/private/tmp/bb_sympy")

import sympy as s

n = s.symbols("n", integer=True, positive=True)


def q(k):
    return (9 * k**2 - 30 * k + 20) / s.Integer(4) ** k


def w(k):
    return -s.Rational(2, 3) * k * (2 * k - 5) * (3 * k - 5) / s.Integer(4) ** k


A = s.Rational(3, 20) * (n + 3) / (n + 1)
B = s.Rational(3, 20) * (n + 4) / n
C = s.Rational(1, 20) * (n + 3) * (n + 4) / (n * (n + 1))

q_residual = s.factor(
    q(n + 5) - (A * q(1) * q(n + 4) - B * q(2) * q(n + 3) + C * q(3) * q(n + 2))
)
w_residual = s.factor(
    w(n + 5)
    - (
        A * (w(1) * q(n + 4) + q(1) * w(n + 4))
        - B * (w(2) * q(n + 3) + q(2) * w(n + 3))
        + C * (w(3) * q(n + 2) + q(3) * w(n + 2))
    )
)

x = s.Rational(1, 4)
infinite_symbol = s.simplify(6 * x * (1 + x) / (1 - x) ** 3 - 10 * x / (1 - x) ** 2)

print("q recurrence residual:", q_residual)
print("w recurrence residual:", w_residual)
print("infinite vector symbol:", infinite_symbol)

raise SystemExit(0 if q_residual == w_residual == infinite_symbol == 0 else 1)