def test_rsolve_hyper():
assert rsolve_hyper([-1, -1, 1], 0, n) in [
C0 * (S.Half - S.Half * sqrt(5)) ** n + C1 * (S.Half + S.Half * sqrt(5)) ** n,
C1 * (S.Half - S.Half * sqrt(5)) ** n + C0 * (S.Half + S.Half * sqrt(5)) ** n,
]
assert rsolve_hyper([n ** 2 - 2, -2 * n - 1, 1], 0, n) in [
C0 * rf(sqrt(2), n) + C1 * rf(-sqrt(2), n),
C1 * rf(sqrt(2), n) + C0 * rf(-sqrt(2), n),
]
assert rsolve_hyper([n ** 2 - k, -2 * n - 1, 1], 0, n) in [
C0 * rf(sqrt(k), n) + C1 * rf(-sqrt(k), n),
C1 * rf(sqrt(k), n) + C0 * rf(-sqrt(k), n),
]
assert rsolve_hyper([2 * n * (n + 1), -n ** 2 - 3 * n + 2, n - 1], 0, n) == C1 * factorial(n) + C0 * 2 ** n
assert rsolve_hyper([n + 2, -(2 * n + 3) * (17 * n ** 2 + 51 * n + 39), n + 1], 0, n) == 0
assert rsolve_hyper([-n - 1, -1, 1], 0, n) == 0
assert rsolve_hyper([-1, 1], n, n).expand() == C0 + n ** 2 / 2 - n / 2
assert rsolve_hyper([-1, 1], 1 + n, n).expand() == C0 + n ** 2 / 2 + n / 2
assert rsolve_hyper([-1, 1], 3 * (n + n ** 2), n).expand() == C0 + n ** 3 - n
def test_rsolve_hyper():
assert rsolve_hyper(
[-1, -1, 1], 0,
n) == C0 * (S.Half - S.Half * sqrt(5))**n + C1 * (S.Half +
S.Half * sqrt(5))**n
assert rsolve_hyper([n**2 - 2, -2 * n - 1, 1], 0, n) in [
C0 * rf(sqrt(2), n) + C1 * rf(-sqrt(2), n),
C1 * rf(sqrt(2), n) + C0 * rf(-sqrt(2), n)
]
assert rsolve_hyper([n**2 - k, -2 * n - 1, 1], 0, n) in [
C0 * rf(sqrt(k), n) + C1 * rf(-sqrt(k), n),
C1 * rf(sqrt(k), n) + C0 * rf(-sqrt(k), n)
]
assert rsolve_hyper([2 * n * (n + 1), -n**2 - 3 * n + 2, n - 1], 0,
n) == C0 * factorial(n) + C1 * 2**n
assert rsolve_hyper(
[n + 2, -(2 * n + 3) * (17 * n**2 + 51 * n + 39), n + 1], 0, n) == 0
assert rsolve_hyper([-n - 1, -1, 1], 0, n) == 0
assert rsolve_hyper([-1, 1], n, n).expand() == C0 + n**2 / 2 - n / 2
assert rsolve_hyper([-1, 1], 1 + n, n).expand() == C0 + n**2 / 2 + n / 2
assert rsolve_hyper([-1, 1], 3 * (n + n**2), n).expand() == C0 + n**3 - n
def test_rsolve_hyper():
assert rsolve_hyper([-1, -1, 1], 0, n) in [
C0 * (S.Half - S.Half * sqrt(5))**n + C1 *
(S.Half + S.Half * sqrt(5))**n,
C1 * (S.Half - S.Half * sqrt(5))**n + C0 *
(S.Half + S.Half * sqrt(5))**n,
]
assert rsolve_hyper([n**2 - 2, -2 * n - 1, 1], 0, n) in [
C0 * rf(sqrt(2), n) + C1 * rf(-sqrt(2), n),
C1 * rf(sqrt(2), n) + C0 * rf(-sqrt(2), n),
]
assert rsolve_hyper([n**2 - k, -2 * n - 1, 1], 0, n) in [
C0 * rf(sqrt(k), n) + C1 * rf(-sqrt(k), n),
C1 * rf(sqrt(k), n) + C0 * rf(-sqrt(k), n),
]
assert rsolve_hyper([2 * n * (n + 1), -n**2 - 3 * n + 2, n - 1], 0,
n) == C1 * factorial(n) + C0 * 2**n
assert rsolve_hyper(
[n + 2, -(2 * n + 3) * (17 * n**2 + 51 * n + 39), n + 1], 0, n) == None
assert rsolve_hyper([-n - 1, -1, 1], 0, n) == None
assert rsolve_hyper([-1, 1], n, n).expand() == C0 + n**2 / 2 - n / 2
assert rsolve_hyper([-1, 1], 1 + n, n).expand() == C0 + n**2 / 2 + n / 2
assert rsolve_hyper([-1, 1], 3 * (n + n**2), n).expand() == C0 + n**3 - n
assert rsolve_hyper([-a, 1], 0, n).expand() == C0 * a**n
assert rsolve_hyper(
[-a, 0, 1], 0,
n).expand() == (-1)**n * C1 * a**(n / 2) + C0 * a**(n / 2)
assert rsolve_hyper([1, 1, 1], 0, n).expand() == \
C0*(-S(1)/2 - sqrt(3)*I/2)**n + C1*(-S(1)/2 + sqrt(3)*I/2)**n
assert rsolve_hyper([1, -2 * n / a - 2 / a, 1], 0, n) == None
def test_rsolve_hyper():
assert rsolve_hyper([-1, -1, 1], 0, n) in [
C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n,
C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n,
]
assert rsolve_hyper([n**2 - 2, -2*n - 1, 1], 0, n) in [
C0*rf(sqrt(2), n) + C1*rf(-sqrt(2), n),
C1*rf(sqrt(2), n) + C0*rf(-sqrt(2), n),
]
assert rsolve_hyper([n**2 - k, -2*n - 1, 1], 0, n) in [
C0*rf(sqrt(k), n) + C1*rf(-sqrt(k), n),
C1*rf(sqrt(k), n) + C0*rf(-sqrt(k), n),
]
assert rsolve_hyper(
[2*n*(n + 1), -n**2 - 3*n + 2, n - 1], 0, n) == C1*factorial(n) + C0*2**n
assert rsolve_hyper(
[n + 2, -(2*n + 3)*(17*n**2 + 51*n + 39), n + 1], 0, n) == None
assert rsolve_hyper([-n - 1, -1, 1], 0, n) == None
assert rsolve_hyper([-1, 1], n, n).expand() == C0 + n**2/2 - n/2
assert rsolve_hyper([-1, 1], 1 + n, n).expand() == C0 + n**2/2 + n/2
assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n
assert rsolve_hyper([-a, 1],0,n).expand() == C0*a**n
assert rsolve_hyper([-a, 0, 1], 0, n).expand() == (-1)**n*C1*a**(n/2) + C0*a**(n/2)
assert rsolve_hyper([1, 1, 1], 0, n).expand() == \
C0*(-S(1)/2 - sqrt(3)*I/2)**n + C1*(-S(1)/2 + sqrt(3)*I/2)**n
assert rsolve_hyper([1, -2*n/a - 2/a, 1], 0, n) == None
