OFFSET
0,1
COMMENTS
Positive root of the minimal polynomial x^2 + 1/5 - 1/5. The negative root is -(1/5)*A222134 = -0.558257569...
c^n = A(-n) + B(-n)*phi21, and A(n) = S21(n+1) - S21(n) = A365824(n), with phi21 = A222134, and B(n) = S21(n) = A015440(n-1), where S21(n) = sqrt(-5)^(n-1)*S(n-1, 1/sqrt(-5)), with the Chebyshev polynomials {S(n, x)} (see A049310).
The formula for negative indices of S is S(-1, 0) = 0 and S(-n, x) = -S(n-2, x) for n >= 2.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..10000
FORMULA
c = 1/phi21 = (1/5)*(1 - phi21), with phi21 = (1 + sqrt(21))/2 = A222134, hence an algebraic number of the real quadratic number field Q(sqrt(21)) but not an algebraic integer like phi24.
Equals (A010477-1)/10. - R. J. Mathar, Nov 21 2023
Equals 2*A222135/10. - Hugo Pfoertner, Mar 21 2024
EXAMPLE
c = 0.3582575694955840006588047193728008488984456...
MATHEMATICA
First[RealDigits[(Sqrt[21]-1)/10, 10, 100]] (* Paolo Xausa, Nov 21 2023 *)
PROG
(PARI) \\ Works in v2.13 and higher; n = 100 decimal places
my(n=100); digits(floor(10^(n-1)*(quadgen(84)-1))) \\ Michal Paulovic, Nov 20 2023
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Nov 20 2023
STATUS
approved