A367453 Decimal expansion of (-1 + sqrt(21))/10 = 1/A222134.

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

Offset: 0

Author

Wolfdieter Lang, Nov 20 2023

Keywords

• nonn
• cons
• easy

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.

Examples

			c = 0.3582575694955840006588047193728008488984456...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A010477, A049310, A222134, A222135, A365824.

Programs

Mathematica

First[RealDigits[(Sqrt[21]-1)/10,10,100]] (* Paolo Xausa, Nov 21 2023 *)

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

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