A395973 Decimal expansion of the solution of phi^(2*x)*cos(Pi*x) = (27 + 7*sqrt(5))/22, where phi = (1 + sqrt(5)) / 2.

1, 6, 3, 2, 0, 5, 1, 6, 6, 9, 8, 4, 7, 5, 7, 7, 4, 8, 7, 6, 7, 4, 0, 6, 2, 6, 1, 7, 2, 1, 3, 5, 7, 7, 3, 9, 9, 0, 6, 1, 8, 5, 9, 0, 6, 7, 5, 4, 2, 3, 3, 0, 3, 4, 6, 3, 0, 9, 1, 0, 9, 5, 4, 1, 3, 5, 8, 0, 2, 4, 2, 8, 7, 4, 0, 2, 5, 2, 2, 2, 9, 8, 2, 7, 0, 8, 5, 0

Offset

1,2

Comments

The first real root of the function defining A396201. See the illustration given there. The root is almost certainly transcendental.

Links

Table of n, a(n) for n=1..88.

Example

1.632051669847577487674062617213577399061859...

Prog

(Python)
import mpmath
mpmath.mp.dps = 105
root = guess = 1.63
phi = (1 + mpmath.sqrt(5)) / 2
rhs = (27 + 7 * mpmath.sqrt(5)) / 22
f = lambda x: phi**(2*x) * mpmath.cos(mpmath.pi * x) - rhs
try:
    root = mpmath.findroot(f, guess)
except ValueError as e:
    root = 0
root_str = str(root)
", ".join(char for char in root_str if char.isdigit())

Crossrefs

Cf. A001622, A396201.

Sequence in context: A108451 A122178 A126445 * A277435 A033326 A068996

Adjacent sequences: A395970 A395971 A395972 * A395974 A395975 A395976

Keyword

nonn,cons

Author

Peter Luschny, May 27 2026

Status

approved