def test_diofantissue_294():
f = y(n) - y(n - 1) - 2*y(n - 2) - 2*n
assert rsolve(f, y(n)) == (-1)**n*C0 + 2**n*C1 - n - Rational(5, 2)
# issue sympy/sympy#11261
assert rsolve(f, y(n), {y(0): -1, y(1): 1}) == (-(-1)**n/2 + 2*2**n -
n - Rational(5, 2))
# issue sympy/sympy#7055
assert rsolve(-2*y(n) + y(n + 1) + n - 1, y(n)) == 2**n*C0 + n
def test_diofantissue_294():
f = y(n) - y(n - 1) - 2 * y(n - 2) - 2 * n
assert rsolve(f, y(n)) == (-1)**n * C0 + 2**n * C1 - n - Rational(5, 2)
# issue sympy/sympy#11261
assert rsolve(f, y(n), {
y(0): -1,
y(1): 1
}) == (-(-1)**n / 2 + 2 * 2**n - n - Rational(5, 2))
# issue sympy/sympy#7055
assert rsolve(-2 * y(n) + y(n + 1) + n - 1, y(n)) == 2**n * C0 + n
def test_sympyissue_8697():
a = Function('a')
eq = a(n + 3) - a(n + 2) - a(n + 1) + a(n)
assert rsolve(eq, a(n)) == (-1)**n*C1 + C0 + C2*n
eq2 = a(n + 3) + 3*a(n + 2) + 3*a(n + 1) + a(n)
assert (rsolve(eq2, a(n)) ==
(-1)**n*C0 + (-1)**(n + 1)*C1*n + (-1)**(n + 1)*C2*n**2)
assert rsolve(a(n) - 2*a(n - 3) + 5*a(n - 2) - 4*a(n - 1),
a(n), {a(0): 1, a(1): 3, a(2): 8}) == 3*2**n - n - 2
# From issue thread (but not related to the problem, fixed before):
assert rsolve(a(n) - 2*a(n - 1) - n, a(n), {a(0): 1}) == 3*2**n - n - 2
assert (rsolve(a(n + 2) - 5*a(n + 1) + 6*a(n) - n, a(n)) ==
2**n*C0 + 3**n*C1 + n/2 + Rational(3, 4))
def test_sympyissue_8697():
a = Function('a')
eq = a(n + 3) - a(n + 2) - a(n + 1) + a(n)
assert rsolve(eq, a(n)) == (-1)**n * C1 + C0 + C2 * n
eq2 = a(n + 3) + 3 * a(n + 2) + 3 * a(n + 1) + a(n)
assert (rsolve(eq2, a(n)) == (-1)**n * C0 + (-1)**(n + 1) * C1 * n +
(-1)**(n + 1) * C2 * n**2)
assert rsolve(
a(n) - 2 * a(n - 3) + 5 * a(n - 2) - 4 * a(n - 1), a(n), {
a(0): 1,
a(1): 3,
a(2): 8
}) == 3 * 2**n - n - 2
# From issue thread (but not related to the problem, fixed before):
assert rsolve(a(n) - 2 * a(n - 1) - n, a(n), {a(0): 1}) == 3 * 2**n - n - 2
assert (rsolve(a(n + 2) - 5 * a(n + 1) + 6 * a(n) - n,
a(n)) == 2**n * C0 + 3**n * C1 + n / 2 + Rational(3, 4))
def test_rsolve_raises():
x = Function('x')
pytest.raises(ValueError, lambda: rsolve(y(n) - y(k + 1), y(n)))
pytest.raises(NotImplementedError, lambda: rsolve(y(n) - y(n + 1), x(n)))
pytest.raises(NotImplementedError, lambda: rsolve(y(n) - x(n + 1), y(n)))
pytest.raises(ValueError, lambda: rsolve(y(n) - sqrt(n)*y(n + 1), y(n)))
pytest.raises(ValueError, lambda: rsolve(y(n) - y(n + 1), y(n), {x(0): 0}))
pytest.raises(NotImplementedError, lambda: rsolve(y(n) - y(n + 1) + y(n - 1)**2, y(n)))
pytest.raises(NotImplementedError, lambda: rsolve(y(n) - sin(n), y(n)))
# sympy/sympy#11063
pytest.raises(NotImplementedError, lambda: rsolve(y(n + 1, a) - y(n, 2*a),
y(n, a), {y(0, a): a}))
def test_rsolve_raises():
x = Function('x')
pytest.raises(ValueError, lambda: rsolve(y(n) - y(k + 1), y(n)))
pytest.raises(NotImplementedError, lambda: rsolve(y(n) - y(n + 1), x(n)))
pytest.raises(NotImplementedError, lambda: rsolve(y(n) - x(n + 1), y(n)))
pytest.raises(ValueError, lambda: rsolve(y(n) - sqrt(n) * y(n + 1), y(n)))
pytest.raises(ValueError, lambda: rsolve(y(n) - y(n + 1), y(n), {x(0): 0}))
pytest.raises(NotImplementedError,
lambda: rsolve(y(n) - y(n + 1) + y(n - 1)**2, y(n)))
pytest.raises(NotImplementedError, lambda: rsolve(y(n) - sin(n), y(n)))
# sympy/sympy#11063
pytest.raises(
NotImplementedError,
lambda: rsolve(y(n + 1, a) - y(n, 2 * a), y(n, a), {y(0, a): a}))
def test_diofantissue_451():
assert rsolve(y(n) - 2 * y(n - 1) - 3**n, y(n),
{y(0): 1}) == 3**(n + 1) - 2 * 2**n
def test_sympyissue_6844():
f = y(n + 2) - y(n + 1) + y(n) / 4
assert rsolve(f, y(n)) == 2**(-n + 1) * C1 * n + 2**(-n) * C0
assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2**(1 - n) * n
def test_rsolve():
f = y(n + 2) - y(n + 1) - y(n)
h = sqrt(5)*(Rational(1, 2) + sqrt(5)/2)**n \
- sqrt(5)*(Rational(1, 2) - sqrt(5)/2)**n
assert rsolve(f, y(n)) in [
C0 * (Rational(1, 2) - sqrt(5) / 2)**n + C1 *
(Rational(1, 2) + sqrt(5) / 2)**n,
C1 * (Rational(1, 2) - sqrt(5) / 2)**n + C0 *
(Rational(1, 2) + sqrt(5) / 2)**n,
]
assert rsolve(f, y(n), [0, 5]) == h
assert rsolve(f, y(n), {0: 0, 1: 5}) == h
assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h
assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h
assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = (n - 1) * y(n + 2) - (n**2 + 3 * n - 2) * y(n + 1) + 2 * n * (n +
1) * y(n)
g = C1 * factorial(n) + C0 * 2**n
h = -3 * factorial(n) + 3 * 2**n
assert rsolve(f, y(n)) == g
assert rsolve(f, y(n), []) == g
assert rsolve(f, y(n), {}) == g
assert rsolve(f, y(n), [0, 3]) == h
assert rsolve(f, y(n), {0: 0, 1: 3}) == h
assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = y(n) - y(n - 1) - 2
assert rsolve(f, y(n), {y(0): 0}) == 2 * n
assert rsolve(f, y(n), {y(0): 1}) == 2 * n + 1
assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = 3 * y(n - 1) - y(n) - 1
assert rsolve(f, y(n), {y(0): 0}) == -3**n / 2 + Rational(1, 2)
assert rsolve(f, y(n), {y(0): 1}) == 3**n / 2 + Rational(1, 2)
assert rsolve(f, y(n), {y(0): 2}) == 3 * 3**n / 2 + Rational(1, 2)
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = y(n) - 1 / n * y(n - 1)
assert rsolve(f, y(n)) == C0 / factorial(n)
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = y(n) - 1 / n * y(n - 1) - 1
assert rsolve(f, y(n)) is None
f = 2 * y(n - 1) + (1 - n) * y(n) / n
assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1) * n
assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1) * n * 2
assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1) * n * 3
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = (n - 1) * (n - 2) * y(n + 2) - (n + 1) * (n + 2) * y(n)
assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n * (n - 1) * (n - 2)
assert rsolve(f, y(n), {
y(3): 6,
y(4): -24
}) == -n * (n - 1) * (n - 2) * (-1)**(n)
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
assert rsolve(Eq(y(n + 1), a * y(n)), y(n), {y(1): a}).simplify() == a**n
assert rsolve(y(n) - a*y(n-2), y(n),
{y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \
a**(n/2)*(-(-1)**n*b + a)
f = (-16 * n**2 + 32 * n - 12) * y(n - 1) + (4 * n**2 - 12 * n + 9) * y(n)
assert expand_func(rsolve(f, y(n),
{y(1): binomial(2*n + 1, 3)}).rewrite(gamma)).simplify() == \
2**(2*n)*n*(2*n - 1)*(4*n**2 - 1)/12
assert (rsolve(y(n) + a * (y(n + 1) + y(n - 1)) / 2, y(n)) -
(C0 * (-sqrt(-1 + a**(-2)) - 1 / a)**n + C1 *
(sqrt(-1 + a**(-2)) - 1 / a)**n)).simplify() == 0
def test_rsolve():
f = y(n + 2) - y(n + 1) - y(n)
h = sqrt(5)*(Rational(1, 2) + sqrt(5)/2)**n \
- sqrt(5)*(Rational(1, 2) - sqrt(5)/2)**n
assert rsolve(f, y(n)) in [
C0*(Rational(1, 2) - sqrt(5)/2)**n + C1*(Rational(1, 2) + sqrt(5)/2)**n,
C1*(Rational(1, 2) - sqrt(5)/2)**n + C0*(Rational(1, 2) + sqrt(5)/2)**n,
]
assert rsolve(f, y(n), [0, 5]) == h
assert rsolve(f, y(n), {0: 0, 1: 5}) == h
assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h
assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h
assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h
assert f.subs({y: Lambda(k, rsolve(f, y(n)).subs({n: k}))}).simplify() == 0
f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n)
g = C1*factorial(n) + C0*2**n
h = -3*factorial(n) + 3*2**n
assert rsolve(f, y(n)) == g
assert rsolve(f, y(n), []) == g
assert rsolve(f, y(n), {}) == g
assert rsolve(f, y(n), [0, 3]) == h
assert rsolve(f, y(n), {0: 0, 1: 3}) == h
assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h
assert f.subs({y: Lambda(k, rsolve(f, y(n)).subs({n: k}))}).simplify() == 0
f = y(n) - y(n - 1) - 2
assert rsolve(f, y(n), {y(0): 0}) == 2*n
assert rsolve(f, y(n), {y(0): 1}) == 2*n + 1
assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None
assert f.subs({y: Lambda(k, rsolve(f, y(n)).subs({n: k}))}).simplify() == 0
f = 3*y(n - 1) - y(n) - 1
assert rsolve(f, y(n), {y(0): 0}) == -3**n/2 + Rational(1, 2)
assert rsolve(f, y(n), {y(0): 1}) == 3**n/2 + Rational(1, 2)
assert rsolve(f, y(n), {y(0): 2}) == 3*3**n/2 + Rational(1, 2)
assert f.subs({y: Lambda(k, rsolve(f, y(n)).subs({n: k}))}).simplify() == 0
f = y(n) - 1/n*y(n - 1)
assert rsolve(f, y(n)) == C0/factorial(n)
assert f.subs({y: Lambda(k, rsolve(f, y(n)).subs({n: k}))}).simplify() == 0
f = y(n) - 1/n*y(n - 1) - 1
assert rsolve(f, y(n)) is None
f = 2*y(n - 1) + (1 - n)*y(n)/n
assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1)*n
assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1)*n*2
assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1)*n*3
assert f.subs({y: Lambda(k, rsolve(f, y(n)).subs({n: k}))}).simplify() == 0
f = (n - 1)*(n - 2)*y(n + 2) - (n + 1)*(n + 2)*y(n)
assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n*(n - 1)*(n - 2)
assert rsolve(
f, y(n), {y(3): 6, y(4): -24}) == -n*(n - 1)*(n - 2)*(-1)**(n)
assert f.subs({y: Lambda(k, rsolve(f, y(n)).subs({n: k}))}).simplify() == 0
assert rsolve(Eq(y(n + 1), a*y(n)), y(n), {y(1): a}).simplify() == a**n
assert rsolve(y(n) - a*y(n-2), y(n),
{y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \
a**(n/2)*(-(-1)**n*b + a)
f = (-16*n**2 + 32*n - 12)*y(n - 1) + (4*n**2 - 12*n + 9)*y(n)
assert expand_func(rsolve(f, y(n),
{y(1): binomial(2*n + 1, 3)}).rewrite(gamma)).simplify() == \
2**(2*n)*n*(2*n - 1)*(4*n**2 - 1)/12
assert (rsolve(y(n) + a*(y(n + 1) + y(n - 1))/2, y(n)) -
(C0*(-sqrt(-1 + a**(-2)) - 1/a)**n +
C1*(sqrt(-1 + a**(-2)) - 1/a)**n)).simplify() == 0
def test_diofantissue_13629():
assert rsolve(y(n + 1) - (y(n) + (n + 1)**2),
y(n), {y(0): 0}) == n*(2*n**2 + 3*n + 1)/6
def test_diofantissue_456():
assert rsolve(y(n) - 2*y(n - 1) - 3**n*n,
y(n), {y(0): 1}) == 7*2**n + 3**(n + 1)*(n - 2)
def test_diofantissue_451():
assert rsolve(y(n) - 2*y(n - 1) - 3**n, y(n),
{y(0): 1}) == 3**(n+1) - 2*2**n
def test_sympyissue_6844():
f = y(n + 2) - y(n + 1) + y(n)/4
assert rsolve(f, y(n)) == 2**(-n + 1)*C1*n + 2**(-n)*C0
assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2**(1 - n)*n
def test_diofantissue_456():
assert rsolve(y(n) - 2 * y(n - 1) - 3**n * n, y(n),
{y(0): 1}) == 7 * 2**n + 3**(n + 1) * (n - 2)
def test_issue_6844():
f = y(n + 2) - y(n + 1) + y(n) / 4
assert rsolve(f, y(n)) == 2**(-n) * (C0 + C1 * n)
assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2 * 2**(-n) * n
def test_diofantissue_13629():
assert rsolve(y(n + 1) - (y(n) + (n + 1)**2), y(n),
{y(0): 0}) == n * (2 * n**2 + 3 * n + 1) / 6
def test_sympyissue_15553():
assert rsolve(Eq(y(n + 1), 2 * y(n) + n**2 + 1),
y(n)) == 2**n * C0 - n**2 - 2 * n - 4
assert rsolve(Eq(y(n + 1), 2 * y(n) + n**2 + 1), y(n),
{y(1): 0}) == 7 * 2**n / 2 - n**2 - 2 * n - 4
